{"id":1848,"date":"2018-01-11T20:54:31","date_gmt":"2018-01-11T20:54:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/the-chain-rule\/"},"modified":"2018-01-31T20:48:03","modified_gmt":"2018-01-31T20:48:03","slug":"the-chain-rule","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/chapter\/the-chain-rule\/","title":{"raw":"3.6 The Chain Rule","rendered":"3.6 The Chain Rule"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>State the chain rule for the composition of two functions.<\/li>\r\n \t<li>Apply the chain rule together with the power rule.<\/li>\r\n \t<li>Apply the chain rule and the product\/quotient rules correctly in combination when both are necessary.<\/li>\r\n \t<li>Recognize the chain rule for a composition of three or more functions.<\/li>\r\n \t<li>Describe the proof of the chain rule.<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe have seen the techniques for differentiating basic functions [latex]({x}^{n}, \\sin x, \\cos x,\\text{etc}.)[\/latex] as well as sums, differences, products, quotients, and constant multiples of these functions. However, these techniques do not allow us to differentiate compositions of functions, such as [latex]h(x)= \\sin ({x}^{3})[\/latex] or [latex]k(x)=\\sqrt{3{x}^{2}+1}.[\/latex] In this section, we study the rule for finding the derivative of the composition of two or more functions.\r\n<div id=\"fs-id1169738923754\" class=\"bc-section section\">\r\n<h1>Deriving the Chain Rule<\/h1>\r\n<p id=\"fs-id1169738824611\">When we have a function that is a composition of two or more functions, we could use all of the techniques we have already learned to differentiate it. However, using all of those techniques to break down a function into simpler parts that we are able to differentiate can get cumbersome. Instead, we use the <strong>chain rule<\/strong>, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.<\/p>\r\n<p id=\"fs-id1169736656600\">To put this rule into context, let\u2019s take a look at an example: [latex]h(x)= \\sin ({x}^{3}).[\/latex] We can think of the derivative of this function with respect to [latex]x[\/latex] as the rate of change of [latex] \\sin ({x}^{3})[\/latex] relative to the change in [latex]x.[\/latex] Consequently, we want to know how [latex] \\sin ({x}^{3})[\/latex] changes as [latex]x[\/latex] changes. We can think of this event as a chain reaction: As [latex]x[\/latex] changes, [latex]{x}^{3}[\/latex] changes, which leads to a change in [latex] \\sin ({x}^{3}).[\/latex] This chain reaction gives us hints as to what is involved in computing the derivative of [latex] \\sin ({x}^{3}).[\/latex] First of all, a change in [latex]x[\/latex] forcing a change in [latex]{x}^{3}[\/latex] suggests that somehow the derivative of [latex]{x}^{3}[\/latex] is involved. In addition, the change in [latex]{x}^{3}[\/latex] forcing a change in [latex] \\sin ({x}^{3})[\/latex] suggests that the derivative of [latex] \\sin (u)[\/latex] with respect to [latex]u,[\/latex] where [latex]u={x}^{3},[\/latex] is also part of the final derivative.<\/p>\r\n<p id=\"fs-id1169738823074\">We can take a more formal look at the derivative of [latex]h(x)= \\sin ({x}^{3})[\/latex] by setting up the limit that would give us the derivative at a specific value [latex]a[\/latex] in the domain of [latex]h(x)= \\sin ({x}^{3}).[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169738941291\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(a)=\\underset{x\\to a}{\\text{lim}}\\frac{ \\sin ({x}^{3})- \\sin ({a}^{3})}{x-a}.[\/latex]<\/div>\r\n<p id=\"fs-id1169738954231\">This expression does not seem particularly helpful; however, we can modify it by multiplying and dividing by the expression [latex]{x}^{3}-{a}^{3}[\/latex] to obtain<\/p>\r\n\r\n<div id=\"fs-id1169739101346\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(a)=\\underset{x\\to a}{\\text{lim}}\\frac{ \\sin ({x}^{3})- \\sin ({a}^{3})}{{x}^{3}-{a}^{3}}\u00b7\\frac{{x}^{3}-{a}^{3}}{x-a}.[\/latex]<\/div>\r\n<p id=\"fs-id1169738969849\">From the definition of the derivative, we can see that the second factor is the derivative of [latex]{x}^{3}[\/latex] at [latex]x=a.[\/latex] That is,<\/p>\r\n\r\n<div id=\"fs-id1169739096283\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\text{lim}}\\frac{{x}^{3}-{a}^{3}}{x-a}=\\frac{d}{dx}({x}^{3})=3{a}^{2}.[\/latex]<\/div>\r\n<p id=\"fs-id1169738971694\">However, it might be a little more challenging to recognize that the first term is also a derivative. We can see this by letting [latex]u={x}^{3}[\/latex] and observing that as [latex]x\\to a,u\\to {a}^{3}\\text{:}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169739035291\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\underset{x\\to a}{\\text{lim}}\\frac{ \\sin ({x}^{3})- \\sin ({a}^{3})}{{x}^{3}-{a}^{3}}&amp; =\\underset{u\\to {a}^{3}}{\\text{lim}}\\frac{ \\sin u- \\sin ({a}^{3})}{u-{a}^{3}}\\hfill \\\\ &amp; =\\frac{d}{du}{( \\sin u)}_{u={a}^{3}}\\hfill \\\\ &amp; = \\cos ({a}^{3}).\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1169739000166\">Thus, [latex]{h}^{\\prime }(a)= \\cos ({a}^{3})\u00b73{a}^{2}.[\/latex]<\/p>\r\n<p id=\"fs-id1169739044550\">In other words, if [latex]h(x)= \\sin ({x}^{3}),[\/latex] then [latex]{h}^{\\prime }(x)= \\cos ({x}^{3})\u00b73{x}^{2}.[\/latex] Thus, if we think of [latex]h(x)= \\sin ({x}^{3})[\/latex] as the composition [latex](f\\circ g)(x)=f(g(x))[\/latex] where [latex]f(x)=[\/latex] sin [latex]x[\/latex] and [latex]g(x)={x}^{3},[\/latex] then the derivative of [latex]h(x)= \\sin ({x}^{3})[\/latex] is the product of the derivative of [latex]g(x)={x}^{3}[\/latex] and the derivative of the function [latex]f(x)= \\sin x[\/latex] evaluated at the function [latex]g(x)={x}^{3}.[\/latex] At this point, we anticipate that for [latex]h(x)= \\sin (g(x)),[\/latex] it is quite likely that [latex]{h}^{\\prime }(x)= \\cos (g(x)){g}^{\\prime }(x).[\/latex] As we determined above, this is the case for [latex]h(x)= \\sin ({x}^{3}).[\/latex]<\/p>\r\n<p id=\"fs-id1169738955406\">Now that we have derived a special case of the chain rule, we state the general case and then apply it in a general form to other composite functions. An informal proof is provided at the end of the section.<\/p>\r\n\r\n<div id=\"fs-id1169739194586\" class=\"textbox key-takeaways\">\r\n<h3>Rule: The Chain Rule<\/h3>\r\n<p id=\"fs-id1169736612521\">Let [latex]f[\/latex] and [latex]g[\/latex] be functions. For all [latex]x[\/latex] in the domain of [latex]g[\/latex] for which [latex]g[\/latex] is differentiable at [latex]x[\/latex] and [latex]f[\/latex] is differentiable at [latex]g(x),[\/latex] the derivative of the composite function<\/p>\r\n\r\n<div id=\"fs-id1169739233799\" class=\"equation unnumbered\">[latex]h(x)=(f\\circ g)(x)=f(g(x))[\/latex]<\/div>\r\n<p id=\"fs-id1169738949353\">is given by<\/p>\r\n\r\n<div id=\"fs-id1169738948938\" class=\"equation\">[latex]{h}^{\\prime }(x)={f}^{\\prime }(g(x)){g}^{\\prime }(x).[\/latex]<\/div>\r\n<p id=\"fs-id1169739199751\">Alternatively, if [latex]y[\/latex] is a function of [latex]u,[\/latex] and [latex]u[\/latex] is a function of [latex]x,[\/latex] then<\/p>\r\n\r\n<div id=\"fs-id1169739187558\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=\\frac{dy}{du}\u00b7\\frac{du}{dx}.[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738971083\" class=\"textbox tryit media-2\">\r\n<p id=\"fs-id1169739080685\">Watch an <a href=\"http:\/\/www.openstaxcollege.org\/l\/20_chainrule2\">animation<\/a> of the chain rule.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736619728\" class=\"textbox key-takeaways problem-solving\">\r\n<h3>Problem-Solving Strategy: Applying the Chain Rule<\/h3>\r\n<ol id=\"fs-id1169739182671\">\r\n \t<li>To differentiate [latex]h(x)=f(g(x)),[\/latex] begin by identifying [latex]f(x)[\/latex] and [latex]g(x).[\/latex]<\/li>\r\n \t<li>Find [latex]f\\prime (x)[\/latex] and evaluate it at [latex]g(x)[\/latex] to obtain [latex]{f}^{\\prime }(g(x)).[\/latex]<\/li>\r\n \t<li>Find [latex]{g}^{\\prime }(x).[\/latex]<\/li>\r\n \t<li>Write [latex]{h}^{\\prime }(x)={f}^{\\prime }(g(x))\u00b7{g}^{\\prime }(x).[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1169739269865\"><em>Note<\/em>: When applying the chain rule to the composition of two or more functions, keep in mind that we work our way from the outside function in. It is also useful to remember that the derivative of the composition of two functions can be thought of as having two parts; the derivative of the composition of three functions has three parts; and so on. Also, remember that we never evaluate a derivative at a derivative.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739001005\" class=\"bc-section section\">\r\n<h1>The Chain and Power Rules Combined<\/h1>\r\n<p id=\"fs-id1169739096228\">We can now apply the chain rule to composite functions, but note that we often need to use it with other rules. For example, to find derivatives of functions of the form [latex]h(x)={(g(x))}^{n},[\/latex] we need to use the chain rule combined with the power rule. To do so, we can think of [latex]h(x)={(g(x))}^{n}[\/latex] as [latex]f(g(x))[\/latex] where [latex]f(x)={x}^{n}.[\/latex] Then [latex]{f}^{\\prime }(x)=n{x}^{n-1}.[\/latex] Thus, [latex]{f}^{\\prime }(g(x))=n{(g(x))}^{n-1}.[\/latex] This leads us to the derivative of a power function using the chain rule,<\/p>\r\n\r\n<div id=\"fs-id1169739325710\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)=n{(g(x))}^{n-1}{g}^{\\prime }(x)[\/latex]<\/div>\r\n<div id=\"fs-id1169739187734\" class=\"textbox key-takeaways\">\r\n<h3>Rule: Power Rule for Composition of Functions<\/h3>\r\n<p id=\"fs-id1169738978364\">For all values of [latex]x[\/latex] for which the derivative is defined, if<\/p>\r\n\r\n<div id=\"fs-id1169739006308\" class=\"equation unnumbered\">[latex]h(x)={(g(x))}^{n}.[\/latex]<\/div>\r\n<p id=\"fs-id1169739242349\">Then<\/p>\r\n\r\n<div id=\"fs-id1169739222795\" class=\"equation\">[latex]{h}^{\\prime }(x)=n{(g(x))}^{n-1}{g}^{\\prime }(x).[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739274312\" class=\"textbox examples\">\r\n<h3>Using the Chain and Power Rules<\/h3>\r\n<div id=\"fs-id1169739274625\" class=\"exercise\">\r\n<div id=\"fs-id1169739282636\" class=\"textbox\">\r\n<p id=\"fs-id1169736589119\">Find the derivative of [latex]h(x)=\\frac{1}{{(3{x}^{2}+1)}^{2}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736658840\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736658840\"]\r\n<p id=\"fs-id1169736658840\">First, rewrite [latex]h(x)=\\frac{1}{{(3{x}^{2}+1)}^{2}}={(3{x}^{2}+1)}^{-2}.[\/latex]<\/p>\r\n<p id=\"fs-id1169739333152\">Applying the power rule with [latex]g(x)=3{x}^{2}+1,[\/latex] we have<\/p>\r\n\r\n<div id=\"fs-id1169736609881\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)=-2{(3{x}^{2}+1)}^{-3}(6x).[\/latex]<\/div>\r\n<p id=\"fs-id1169736655793\">Rewriting back to the original form gives us<\/p>\r\n\r\n<div id=\"fs-id1169739008104\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)=\\frac{-12x}{{(3{x}^{2}+1)}^{3}}.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736662938\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169736662942\" class=\"exercise\">\r\n<div id=\"fs-id1169736595959\" class=\"textbox\">\r\n<p id=\"fs-id1169736595961\">Find the derivative of [latex]h(x)={(2{x}^{3}+2x-1)}^{4}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739325717\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739325717\"]\r\n<p id=\"fs-id1169739325717\">[latex]{h}^{\\prime }(x)=4{(2{x}^{3}+2x-1)}^{3}(6x+2)=8(3x+1){(2{x}^{3}+2x-1)}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739274677\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169739179049\">Use <a class=\"autogenerated-content\" href=\"#fs-id1169739222795\">(Figure)<\/a> with [latex]g(x)=2{x}^{3}+2x-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739302258\" class=\"textbox examples\">\r\n<h3>Using the Chain and Power Rules with a Trigonometric Function<\/h3>\r\n<div id=\"fs-id1169736609872\" class=\"exercise\">\r\n<div id=\"fs-id1169736609874\" class=\"textbox\">\r\n<p id=\"fs-id1169739273001\">Find the derivative of [latex]h(x)={ \\sin }^{3}x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739182335\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739182335\"]\r\n<p id=\"fs-id1169739182335\">First recall that [latex]{ \\sin }^{3}x={( \\sin x)}^{3},[\/latex] so we can rewrite [latex]h(x)={ \\sin }^{3}x[\/latex] as [latex]h(x)={( \\sin x)}^{3}.[\/latex]<\/p>\r\n<p id=\"fs-id1169739351647\">Applying the power rule with [latex]g(x)= \\sin x,[\/latex] we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739039132\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)=3{( \\sin x)}^{2} \\cos x=3{ \\sin }^{2}x \\cos x.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739190027\" class=\"textbox examples\">\r\n<h3>Finding the Equation of a Tangent Line<\/h3>\r\n<div id=\"fs-id1169739190029\" class=\"exercise\">\r\n<div id=\"fs-id1169736613820\" class=\"textbox\">\r\n<p id=\"fs-id1169736613825\">Find the equation of a line tangent to the graph of [latex]h(x)=\\frac{1}{{(3x-5)}^{2}}[\/latex] at [latex]x=2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739336066\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739336066\"]\r\n<p id=\"fs-id1169739336066\">Because we are finding an equation of a line, we need a point. The [latex]x[\/latex]-coordinate of the point is 2. To find the [latex]y[\/latex]-coordinate, substitute 2 into [latex]h(x).[\/latex] Since [latex]h(2)=\\frac{1}{{(3(2)-5)}^{2}}=1,[\/latex] the point is [latex](2,1).[\/latex]<\/p>\r\n<p id=\"fs-id1169736607681\">For the slope, we need [latex]{h}^{\\prime }(2).[\/latex] To find [latex]{h}^{\\prime }(x),[\/latex] first we rewrite [latex]h(x)={(3x-5)}^{-2}[\/latex] and apply the power rule to obtain<\/p>\r\n\r\n<div id=\"fs-id1169736662505\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)=-2{(3x-5)}^{-3}(3)=-6{(3x-5)}^{-3}.[\/latex]<\/div>\r\n<p id=\"fs-id1169739111347\">By substituting, we have [latex]{h}^{\\prime }(2)=-6{(3(2)-5)}^{-3}=-6.[\/latex] Therefore, the line has equation [latex]y-1=-6(x-2).[\/latex] Rewriting, the equation of the line is [latex]y=-6x+13.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739301868\" class=\"textbox exercises checkpoint\">\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1169739301873\" class=\"textbox\">\r\n<p id=\"fs-id1169739303885\">Find the equation of the line tangent to the graph of [latex]f(x)={({x}^{2}-2)}^{3}[\/latex] at [latex]x=-2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738869705\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738869705\"]\r\n<p id=\"fs-id1169738869705\">[latex]y=-48x-88[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739242781\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169739252012\">Use the preceding example as a guide.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739252019\" class=\"bc-section section\">\r\n<h1>Combining the Chain Rule with Other Rules<\/h1>\r\n<p id=\"fs-id1169739293763\">Now that we can combine the chain rule and the power rule, we examine how to combine the chain rule with the other rules we have learned. In particular, we can use it with the formulas for the derivatives of trigonometric functions or with the product rule.<\/p>\r\n\r\n<div id=\"fs-id1169736656617\" class=\"textbox examples\">\r\n<h3>Using the Chain Rule on a General Cosine Function<\/h3>\r\n<div id=\"fs-id1169736656619\" class=\"exercise\">\r\n<div id=\"fs-id1169736656621\" class=\"textbox\">\r\n<p id=\"fs-id1169739327889\">Find the derivative of [latex]h(x)= \\cos (g(x)).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736657088\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736657088\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736657088\"]Think of [latex]h(x)= \\cos (g(x))[\/latex] as [latex]f(g(x))[\/latex] where [latex]f(x)= \\cos x.[\/latex] Since [latex]{f}^{\\prime }(x)=\\text{\u2212} \\sin x.[\/latex] we have [latex]{f}^{\\prime }(g(x))=\\text{\u2212} \\sin (g(x)).[\/latex] Then we do the following calculation.\r\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill {h}^{\\prime }(x)&amp; ={f}^{\\prime }(g(x)){g}^{\\prime }(x)\\hfill &amp; &amp; &amp; \\text{Apply the chain rule.}\\hfill \\\\ &amp; =\\text{\u2212} \\sin (g(x)){g}^{\\prime }(x)\\hfill &amp; &amp; &amp; \\text{Substitute}{f}^{\\prime }(g(x))=\\text{\u2212} \\sin (g(x)).\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1169739285002\">Thus, the derivative of [latex]h(x)= \\cos (g(x))[\/latex] is given by [latex]{h}^{\\prime }(x)=\\text{\u2212} \\sin (g(x)){g}^{\\prime }(x).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169739301534\">In the following example we apply the rule that we have just derived.<\/p>\r\n\r\n<div id=\"fs-id1169739301537\" class=\"textbox examples\">\r\n<h3>Using the Chain Rule on a Cosine Function<\/h3>\r\n<div id=\"fs-id1169739301539\" class=\"exercise\">\r\n<div id=\"fs-id1169739301541\" class=\"textbox\">\r\n<p id=\"fs-id1169739301547\">Find the derivative of [latex]h(x)= \\cos (5{x}^{2}).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\nLet [latex]g(x)=5{x}^{2}.[\/latex] Then [latex]{g}^{\\prime }(x)=10x.[\/latex] Using the result from the previous example,\r\n<div id=\"fs-id1169739111373\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill {h}^{\\prime }(x)&amp; =\\text{\u2212} \\sin (5{x}^{2})\u00b710x\\hfill \\\\ &amp; =-10x \\sin (5{x}^{2}).\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739333921\" class=\"textbox examples\">\r\n<h3>Using the Chain Rule on Another Trigonometric Function<\/h3>\r\n<div id=\"fs-id1169739333923\" class=\"exercise\">\r\n<div id=\"fs-id1169739333926\" class=\"textbox\">\r\n<p id=\"fs-id1169739333931\">Find the derivative of [latex]h(x)= \\sec (4{x}^{5}+2x).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739300092\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739300092\"]\r\n<p id=\"fs-id1169739300092\">Apply the chain rule to [latex]h(x)= \\sec (g(x))[\/latex] to obtain<\/p>\r\n\r\n<div id=\"fs-id1169739285070\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)= \\sec (g(x) \\tan (g(x)){g}^{\\prime }(x).[\/latex]<\/div>\r\n<p id=\"fs-id1169736615162\">In this problem, [latex]g(x)=4{x}^{5}+2x,[\/latex] so we have [latex]{g}^{\\prime }(x)=20{x}^{4}+2.[\/latex] Therefore, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739299798\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill {h}^{\\prime }(x)&amp; = \\sec (4{x}^{5}+2x) \\tan (4{x}^{5}+2x)(20{x}^{4}+2)\\hfill \\\\ &amp; =(20{x}^{4}+2) \\sec (4{x}^{5}+2x) \\tan (4{x}^{5}+2x).\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739188144\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169739188147\" class=\"exercise\">\r\n<div id=\"fs-id1169739188149\" class=\"textbox\">\r\n<p id=\"fs-id1169739188151\">Find the derivative of [latex]h(x)= \\sin (7x+2).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\n[latex]{h}^{\\prime }(x)=7 \\cos (7x+2)[\/latex]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736607572\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169736607578\">Apply the chain rule to [latex]h(x)= \\sin g(x)[\/latex] first and then use [latex]g(x)=7x+2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169736610159\">At this point we provide a list of derivative formulas that may be obtained by applying the chain rule in conjunction with the formulas for derivatives of trigonometric functions. Their derivations are similar to those used in <a class=\"autogenerated-content\" href=\"#fs-id1169736656617\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1169739333921\">(Figure)<\/a>. For convenience, formulas are also given in Leibniz\u2019s notation, which some students find easier to remember. (We discuss the chain rule using Leibniz\u2019s notation at the end of this section.) It is not absolutely necessary to memorize these as separate formulas as they are all applications of the chain rule to previously learned formulas.<\/p>\r\n\r\n<div id=\"fs-id1169736610174\" class=\"textbox key-takeaways theorem\">\r\n<h3>Using the Chain Rule with Trigonometric Functions<\/h3>\r\n<p id=\"fs-id1169736655841\">For all values of [latex]x[\/latex] for which the derivative is defined,<\/p>\r\n\r\n<div id=\"fs-id1169736655849\" class=\"equation unnumbered\">[latex]\\begin{array}{cccc}\\frac{d}{dx}( \\sin (g(x))= \\cos (g(x))g\\prime (x)\\hfill &amp; &amp; &amp; \\frac{d}{dx} \\sin u= \\cos u\\frac{du}{dx}\\hfill \\\\ \\frac{d}{dx}( \\cos (g(x))=\\text{\u2212} \\sin (g(x))g\\prime (x)\\hfill &amp; &amp; &amp; \\frac{d}{dx} \\cos u=\\text{\u2212} \\sin u\\frac{du}{dx}\\hfill \\\\ \\frac{d}{dx}( \\tan (g(x))\\phantom{\\rule{0.22em}{0ex}}={ \\sec }^{2}(g(x))g\\prime (x)\\hfill &amp; &amp; &amp; \\frac{d}{dx} \\tan u={ \\sec }^{2}u\\frac{du}{dx}\\hfill \\\\ \\frac{d}{dx}( \\cot (g(x))\\phantom{\\rule{0.25em}{0ex}}=\\text{\u2212}{ \\csc }^{2}(g(x))g\\prime (x)\\hfill &amp; &amp; &amp; \\frac{d}{dx} \\cot u\\phantom{\\rule{0.25em}{0ex}}=\\text{\u2212}{ \\csc }^{2}u\\frac{du}{dx}\\hfill \\\\ \\frac{d}{dx}( \\sec (g(x))\\phantom{\\rule{0.25em}{0ex}}= \\sec (g(x) \\tan (g(x))g\\prime (x)\\hfill &amp; &amp; &amp; \\frac{d}{dx} \\sec u\\phantom{\\rule{0.25em}{0ex}}= \\sec u \\tan u\\frac{du}{dx}\\hfill \\\\ \\frac{d}{dx}( \\csc (g(x))\\phantom{\\rule{0.25em}{0ex}}=\\text{\u2212} \\csc (g(x)) \\cot (g(x))g\\prime (x)\\hfill &amp; &amp; &amp; \\frac{d}{dx} \\csc u\\phantom{\\rule{0.25em}{0ex}}=\\text{\u2212} \\csc u \\cot u\\frac{du}{dx}.\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739298047\" class=\"textbox examples\">\r\n<div id=\"fs-id1169739298049\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n<h3>Combining the Chain Rule with the Product Rule<\/h3>\r\n<p id=\"fs-id1169739298056\">Find the derivative of [latex]h(x)={(2x+1)}^{5}{(3x-2)}^{7}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739345782\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739345782\"]\r\n<p id=\"fs-id1169739345782\">First apply the product rule, then apply the chain rule to each term of the product.<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill {h}^{\\prime }(x)&amp; =\\frac{d}{dx}({(2x+1)}^{5})\u00b7{(3x-2)}^{7}+\\frac{d}{dx}({(3x-2)}^{7})\u00b7{(2x+1)}^{5}\\hfill &amp; &amp; &amp; \\text{Apply the product rule.}\\hfill \\\\ &amp; =5{(2x+1)}^{4}\u00b72\u00b7{(3x-2)}^{7}+7{(3x-2)}^{6}\u00b73\u00b7{(2x+1)}^{5}\\hfill &amp; &amp; &amp; \\text{Apply the chain rule.}\\hfill \\\\ &amp; =10{(2x+1)}^{4}{(3x-2)}^{7}+21{(3x-2)}^{6}{(2x+1)}^{5}\\hfill &amp; &amp; &amp; \\text{Simplify.}\\hfill \\\\ &amp; ={(2x+1)}^{4}{(3x-2)}^{6}(10(3x-7)+21(2x+1))\\hfill &amp; &amp; &amp; \\text{Factor out}{(2x+1)}^{4}{(3x-2)}^{6}.\\hfill \\\\ &amp; ={(2x+1)}^{4}{(3x-2)}^{6}(72x-49)\\hfill &amp; &amp; &amp; \\text{Simplify.}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736603462\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169736603465\" class=\"exercise\">\r\n<div id=\"fs-id1169736603467\" class=\"textbox\">\r\n<p id=\"fs-id1169736603469\">Find the derivative of [latex]h(x)=\\frac{x}{{(2x+3)}^{3}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736603515\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736603515\"]\r\n<p id=\"fs-id1169736603515\">[latex]{h}^{\\prime }(x)=\\frac{3-4x}{{(2x+3)}^{4}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736603566\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169736603573\">Start out by applying the quotient rule. Remember to use the chain rule to differentiate the denominator.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736603581\" class=\"bc-section section\">\r\n<h1>Composites of Three or More Functions<\/h1>\r\n<p id=\"fs-id1169739304867\">We can now combine the chain rule with other rules for differentiating functions, but when we are differentiating the composition of three or more functions, we need to apply the chain rule more than once. If we look at this situation in general terms, we can generate a formula, but we do not need to remember it, as we can simply apply the chain rule multiple times.<\/p>\r\n<p id=\"fs-id1169739304874\">In general terms, first we let<\/p>\r\n\r\n<div id=\"fs-id1169739304877\" class=\"equation unnumbered\">[latex]k(x)=h(f(g(x))).[\/latex]<\/div>\r\nThen, applying the chain rule once we obtain\r\n<div id=\"fs-id1169739304927\" class=\"equation unnumbered\">[latex]{k}^{\\prime }(x)=\\frac{d}{dx}(h(f(g(x)))=h\\prime (f(g(x)))\u00b7\\frac{d}{dx}f((g(x))).[\/latex]<\/div>\r\n<p id=\"fs-id1169736597613\">Applying the chain rule again, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169736597616\" class=\"equation unnumbered\">[latex]{k}^{\\prime }(x)={h}^{\\prime }(f(g(x)){f}^{\\prime }(g(x)){g}^{\\prime }(x)).[\/latex]<\/div>\r\n<div id=\"fs-id1169736597703\" class=\"textbox key-takeaways\">\r\n<h3>Rule: Chain Rule for a Composition of Three Functions<\/h3>\r\n<p id=\"fs-id1169736597709\">For all values of [latex]x[\/latex] for which the function is differentiable, if<\/p>\r\n\r\n<div id=\"fs-id1169736597716\" class=\"equation unnumbered\">[latex]k(x)=h(f(g(x))),[\/latex]<\/div>\r\n<p id=\"fs-id1169736597763\">then<\/p>\r\n\r\n<div id=\"fs-id1169736597766\" class=\"equation unnumbered\">[latex]{k}^{\\prime }(x)={h}^{\\prime }(f(g(x))){f}^{\\prime }(g(x)){g}^{\\prime }(x).[\/latex]<\/div>\r\n<p id=\"fs-id1169736658568\">In other words, we are applying the chain rule twice.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1169736658572\">Notice that the derivative of the composition of three functions has three parts. (Similarly, the derivative of the composition of four functions has four parts, and so on.) Also, <em>remember, we can always work from the outside in, taking one derivative at a time.<\/em><\/p>\r\n\r\n<div id=\"fs-id1169736658580\" class=\"textbox examples\">\r\n<h3>Differentiating a Composite of Three Functions<\/h3>\r\n<div id=\"fs-id1169736658582\" class=\"exercise\">\r\n<div id=\"fs-id1169736658584\" class=\"textbox\">\r\n<p id=\"fs-id1169736658589\">Find the derivative of [latex]k(x)={ \\cos }^{4}({7x}^{2}+1).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736658640\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736658640\"]\r\n<p id=\"fs-id1169736658640\">First, rewrite [latex]k(x)[\/latex] as<\/p>\r\n\r\n<div id=\"fs-id1169736658656\" class=\"equation unnumbered\">[latex]k(x)={( \\cos (7{x}^{2}+1))}^{4}.[\/latex]<\/div>\r\n<p id=\"fs-id1169739275169\">Then apply the chain rule several times.<\/p>\r\n\r\n<div id=\"fs-id1169739275173\" class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill {k}^{\\prime }(x)&amp; =4{( \\cos (7{x}^{2}+1))}^{3}(\\frac{d}{dx}( \\cos (7{x}^{2}+1))\\hfill &amp; &amp; &amp; \\text{Apply the chain rule.}\\hfill \\\\ &amp; =4{( \\cos (7{x}^{2}+1))}^{3}(\\text{\u2212} \\sin (7{x}^{2}+1))(\\frac{d}{dx}(7{x}^{2}+1))\\hfill &amp; &amp; &amp; \\text{Apply the chain rule.}\\hfill \\\\ &amp; =4{( \\cos (7{x}^{2}+1))}^{3}(\\text{\u2212} \\sin (7{x}^{2}+1))(14x)\\hfill &amp; &amp; &amp; \\text{Apply the chain rule.}\\hfill \\\\ &amp; =-56x \\sin (7{x}^{2}+1){ \\cos }^{3}(7{x}^{2}+1)\\hfill &amp; &amp; &amp; \\text{Simplify.}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739350744\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169739350747\" class=\"exercise\">\r\n<div id=\"fs-id1169739350749\" class=\"textbox\">\r\n<p id=\"fs-id1169739350751\">Find the derivative of [latex]h(x)={ \\sin }^{6}({x}^{3}).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739350792\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739350792\"]\r\n<p id=\"fs-id1169739350792\">[latex]{h}^{\\prime }(x)=18{x}^{2}{ \\sin }^{5}({x}^{3}) \\cos ({x}^{3})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739350851\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169739350858\">Rewrite [latex]h(x)={ \\sin }^{6}({x}^{3})={( \\sin ({x}^{3}))}^{6}[\/latex] and use <a class=\"autogenerated-content\" href=\"#fs-id1169736658580\">(Figure)<\/a> as a guide.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736593542\" class=\"textbox examples\">\r\n<h3>Using the Chain Rule in a Velocity Problem<\/h3>\r\n<div id=\"fs-id1169736593544\" class=\"exercise\">\r\n<div id=\"fs-id1169736593546\" class=\"textbox\">\r\n<p id=\"fs-id1169736593552\">A particle moves along a coordinate axis. Its position at time [latex]t[\/latex] is given by [latex]s(t)= \\sin (2t)+ \\cos (3t).[\/latex] What is the velocity of the particle at time [latex]t=\\frac{\\pi }{6}?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736593621\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736593621\"]\r\n<p id=\"fs-id1169736593621\">To find [latex]v(t),[\/latex] the velocity of the particle at time [latex]t,[\/latex] we must differentiate [latex]s(t).[\/latex] Thus,<\/p>\r\n\r\n<div id=\"fs-id1169736593661\" class=\"equation unnumbered\">[latex]v(t)={s}^{\\prime }(t)=2 \\cos (2t)-3 \\sin (3t).[\/latex]<\/div>\r\n<p id=\"fs-id1169739266652\">Substituting [latex]t=\\frac{\\pi }{6}[\/latex] into [latex]v(t),[\/latex] we obtain [latex]v(\\frac{\\pi }{6})=-2.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739266711\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169739266714\" class=\"exercise\">\r\n<div id=\"fs-id1169739266717\" class=\"textbox\">\r\n<p id=\"fs-id1169739266719\">A particle moves along a coordinate axis. Its position at time [latex]t[\/latex] is given by [latex]s(t)= \\sin (4t).[\/latex] Find its acceleration at time [latex]t.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739266765\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739266765\"]\r\n<p id=\"fs-id1169739266765\">[latex]a(t)=-16 \\sin (4t)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736655160\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169736655167\">Acceleration is the second derivative of position.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736655174\" class=\"bc-section section\">\r\n<h2>Proof<\/h2>\r\n<p id=\"fs-id1169736655179\">At this point, we present a very informal proof of the chain rule. For simplicity\u2019s sake we ignore certain issues: For example, we assume that [latex]g(x)\\ne g(a)[\/latex] for [latex]x\\ne a[\/latex] in some open interval containing [latex]a.[\/latex] We begin by applying the limit definition of the derivative to the function [latex]h(x)[\/latex] to obtain [latex]{h}^{\\prime }(a)\\text{:}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169736655258\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(a)=\\underset{x\\to a}{\\text{lim}}\\frac{f(g(x))-f(g(a))}{x-a}.[\/latex]<\/div>\r\n<p id=\"fs-id1169739305452\">Rewriting, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739305455\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(a)=\\underset{x\\to a}{\\text{lim}}\\frac{f(g(x))-f(g(a))}{g(x)-g(a)}\u00b7\\frac{g(x)-g(a)}{x-a}.[\/latex]<\/div>\r\n<p id=\"fs-id1169739305587\">Although it is clear that<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\text{lim}}\\frac{g(x)-g(a)}{x-a}={g}^{\\prime }(a),[\/latex]<\/div>\r\n<p id=\"fs-id1169736662280\">it is not obvious that<\/p>\r\n\r\n<div id=\"fs-id1169736662283\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\text{lim}}\\frac{f(g(x))-f(g(a))}{g(x)-g(a)}={f}^{\\prime }(g(a)).[\/latex]<\/div>\r\n<p id=\"fs-id1169736662391\">To see that this is true, first recall that since [latex]g[\/latex] is differentiable at [latex]a,g[\/latex] is also continuous at [latex]a.[\/latex] Thus,<\/p>\r\n\r\n<div id=\"fs-id1169736662414\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\text{lim}}g(x)=g(a).[\/latex]<\/div>\r\n<p id=\"fs-id1169739303393\">Next, make the substitution [latex]y=g(x)[\/latex] and [latex]b=g(a)[\/latex] and use change of variables in the limit to obtain<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\text{lim}}\\frac{f(g(x))-f(g(a))}{g(x)-g(a)}=\\underset{y\\to b}{\\text{lim}}\\frac{f(y)-f(b)}{y-b}={f}^{\\prime }(b)={f}^{\\prime }(g(a)).[\/latex]<\/div>\r\n<p id=\"fs-id1169739335832\">Finally,<\/p>\r\n\r\n<div id=\"fs-id1169739335835\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(a)=\\underset{x\\to a}{\\text{lim}}\\frac{f(g(x))-f(g(a))}{g(x)-g(a)}\u00b7\\frac{g(x)-g(a)}{x-a}={f}^{\\prime }(g(a)){g}^{\\prime }(a).[\/latex]<\/div>\r\n\u25a1\r\n<div id=\"fs-id1169736613849\" class=\"textbox examples\">\r\n<h3>Using the Chain Rule with Functional Values<\/h3>\r\n<div id=\"fs-id1169736613851\" class=\"exercise\">\r\n<div id=\"fs-id1169736613854\" class=\"textbox\">\r\n<p id=\"fs-id1169736613859\">Let [latex]h(x)=f(g(x)).[\/latex] If [latex]g(1)=4,{g}^{\\prime }(1)=3,[\/latex] and [latex]{f}^{\\prime }(4)=7,[\/latex] find [latex]{h}^{\\prime }(1).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736613978\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736613978\"]\r\n<p id=\"fs-id1169736613978\">Use the chain rule, then substitute.<\/p>\r\n\r\n<div id=\"fs-id1169736613982\" class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill {h}^{\\prime }(1)&amp; ={f}^{\\prime }(g(1)){g}^{\\prime }(1)\\hfill &amp; &amp; &amp; \\text{Apply the chain rule.}\\hfill \\\\ &amp; ={f}^{\\prime }(4)\u00b73\\hfill &amp; &amp; &amp; \\text{Substitute}g(1)=4\\text{ and }{g}^{\\prime }(1)=3.\\hfill \\\\ &amp; =7\u00b73\\hfill &amp; &amp; &amp; \\text{Substitute}f\\prime (4)=7.\\hfill \\\\ &amp; =21\\hfill &amp; &amp; &amp; \\text{Simplify.}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739341441\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169739341444\" class=\"exercise\">\r\n<div id=\"fs-id1169739341446\" class=\"textbox\">\r\n<p id=\"fs-id1169739341448\">Given [latex]h(x)=f(g(x)).[\/latex] If [latex]g(2)=-3,{g}^{\\prime }(2)=4,[\/latex] and [latex]{f}^{\\prime }(-3)=7,[\/latex] find [latex]{h}^{\\prime }(2).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739353312\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739353312\"]\r\n<p id=\"fs-id1169739353312\">28<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739353320\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169739353326\">Follow <a class=\"autogenerated-content\" href=\"#fs-id1169736593542\">(Figure)<\/a>.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739353337\" class=\"bc-section section\">\r\n<h1>The Chain Rule Using Leibniz\u2019s Notation<\/h1>\r\n<p id=\"fs-id1169739353342\">As with other derivatives that we have seen, we can express the chain rule using Leibniz\u2019s notation. This notation for the chain rule is used heavily in physics applications.<\/p>\r\n<p id=\"fs-id1169739353347\">[latex]\\text{ For }h(x)=f(g(x)),[\/latex] let [latex]u=g(x)[\/latex] and [latex]y=h(x)=g(u).[\/latex] Thus,<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)=\\frac{dy}{dx},{f}^{\\prime }(g(x))={f}^{\\prime }(u)=\\frac{dy}{du}\\text{ and }{g}^{\\prime }(x)=\\frac{du}{dx}.[\/latex]<\/div>\r\n<p id=\"fs-id1169739307916\">Consequently,<\/p>\r\n\r\n<div id=\"fs-id1169739307919\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}={h}^{\\prime }(x)={f}^{\\prime }(g(x)){g}^{\\prime }(x)=\\frac{dy}{du}\u00b7\\frac{du}{dx}.[\/latex]<\/div>\r\n<div id=\"fs-id1169739308016\" class=\"textbox key-takeaways\">\r\n<h3>Rule: Chain Rule Using Leibniz\u2019s Notation<\/h3>\r\n<p id=\"fs-id1169739264124\">If [latex]y[\/latex] is a function of [latex]u,[\/latex] and [latex]u[\/latex] is a function of [latex]x,[\/latex] then<\/p>\r\n\r\n<div id=\"fs-id1169739264151\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=\\frac{dy}{du}\u00b7\\frac{du}{dx}.[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739264200\" class=\"textbox examples\">\r\n<h3>Taking a Derivative Using Leibniz\u2019s Notation, Example 1<\/h3>\r\n<div id=\"fs-id1169739264202\" class=\"exercise\">\r\n<div id=\"fs-id1169739264204\" class=\"textbox\">\r\n<p id=\"fs-id1169739264209\">Find the derivative of [latex]y={(\\frac{x}{3x+2})}^{5}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739264248\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739264248\"]\r\n<p id=\"fs-id1169739264248\">First, let [latex]u=\\frac{x}{3x+2}.[\/latex] Thus, [latex]y={u}^{5}.[\/latex] Next, find [latex]\\frac{du}{dx}[\/latex] and [latex]\\frac{dy}{du}.[\/latex] Using the quotient rule,<\/p>\r\n\r\n<div id=\"fs-id1169739264320\" class=\"equation unnumbered\">[latex]\\frac{du}{dx}=\\frac{2}{{(3x+2)}^{2}}[\/latex]<\/div>\r\n<p id=\"fs-id1169739289304\">and<\/p>\r\n\r\n<div id=\"fs-id1169739289307\" class=\"equation unnumbered\">[latex]\\frac{dy}{du}=5{u}^{4}.[\/latex]<\/div>\r\n<p id=\"fs-id1169739289337\">Finally, we put it all together.<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill \\frac{dy}{dx}&amp; =\\frac{dy}{du}\u00b7\\frac{du}{dx}\\hfill &amp; &amp; &amp; \\text{Apply the chain rule.}\\hfill \\\\ &amp; =5{u}^{4}\u00b7\\frac{2}{{(3x+2)}^{2}}\\hfill &amp; &amp; &amp; \\text{Substitute}\\frac{dy}{du}=5{u}^{4}\\text{ and }\\frac{du}{dx}=\\frac{2}{{(3x+2)}^{2}}.\\hfill \\\\ &amp; =5{(\\frac{x}{3x+2})}^{4}\u00b7\\frac{2}{{(3x+2)}^{2}}\\hfill &amp; &amp; &amp; \\text{Substitute}u=\\frac{x}{3x+2}.\\hfill \\\\ &amp; =\\frac{10{x}^{4}}{{(3x+2)}^{6}}\\hfill &amp; &amp; &amp; \\text{Simplify.}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1169736592575\">It is important to remember that, when using the Leibniz form of the chain rule, the final answer must be expressed entirely in terms of the original variable given in the problem.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736592583\" class=\"textbox examples\">\r\n<h3>Taking a Derivative Using Leibniz\u2019s Notation, Example 2<\/h3>\r\n<div id=\"fs-id1169736592585\" class=\"exercise\">\r\n<div id=\"fs-id1169736592587\" class=\"textbox\">\r\n<p id=\"fs-id1169736592592\">Find the derivative of [latex]y= \\tan (4{x}^{2}-3x+1).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736661260\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736661260\"]\r\n<p id=\"fs-id1169736661260\">First, let [latex]u=4{x}^{2}-3x+1.[\/latex] Then [latex]y= \\tan u.[\/latex] Next, find [latex]\\frac{du}{dx}[\/latex] and [latex]\\frac{dy}{du}\\text{:}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169736661338\" class=\"equation unnumbered\">[latex]\\frac{du}{dx}=8x-3\\text{ and }\\frac{dy}{du}={ \\sec }^{2}u.[\/latex]<\/div>\r\n<p id=\"fs-id1169736661398\">Finally, we put it all together.<\/p>\r\n\r\n<div id=\"fs-id1169736661401\" class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill \\frac{dy}{dx}&amp; =\\frac{dy}{du}\u00b7\\frac{du}{dx}\\hfill &amp; &amp; &amp; \\text{Apply the chain rule.}\\hfill \\\\ &amp; ={ \\sec }^{2}u\u00b7(8x-3)\\hfill &amp; &amp; &amp; \\text{Use}\\frac{du}{dx}=8x-3\\text{ and }\\frac{dy}{du}={ \\sec }^{2}u.\\hfill \\\\ &amp; ={ \\sec }^{2}(4{x}^{2}-3x+1)\u00b7(8x-3)\\hfill &amp; &amp; &amp; \\text{Substitute}u=4{x}^{2}-3x+1.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739302081\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169739302086\" class=\"exercise\">\r\n<div id=\"fs-id1169739302088\" class=\"textbox\">\r\n<p id=\"fs-id1169739302090\">Use Leibniz\u2019s notation to find the derivative of [latex]y= \\cos ({x}^{3}).[\/latex] Make sure that the final answer is expressed entirely in terms of the variable [latex]x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736594031\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736594031\"]\r\n<p id=\"fs-id1169736594031\">[latex]\\frac{dy}{dx}=-3{x}^{2} \\sin ({x}^{3})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736594073\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169736594079\">Let [latex]u={x}^{3}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736594101\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1169736594108\">\r\n \t<li>The chain rule allows us to differentiate compositions of two or more functions. It states that for [latex]h(x)=f(g(x)),[\/latex]\r\n<div id=\"fs-id1169736594153\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)={f}^{\\prime }(g(x)){g}^{\\prime }(x).[\/latex]<\/div>\r\nIn Leibniz\u2019s notation this rule takes the form\r\n<div id=\"fs-id1169737159954\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=\\frac{dy}{du}\u00b7\\frac{du}{dx}.[\/latex]<\/div><\/li>\r\n \t<li>We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them.<\/li>\r\n \t<li>The chain rule combines with the power rule to form a new rule:\r\n<div id=\"fs-id1169737470447\" class=\"equation unnumbered\">[latex]\\text{ If }h(x)={(g(x))}^{n},\\text{then}{h}^{\\prime }(x)=n{(g(x))}^{n-1}{g}^{\\prime }(x).[\/latex]<\/div><\/li>\r\n \t<li>When applied to the composition of three functions, the chain rule can be expressed as follows: If [latex]h(x)=f(g(k(x))),[\/latex] then [latex]{h}^{\\prime }(x)={f}^{\\prime }(g(k(x)){g}^{\\prime }(k(x)){k}^{\\prime }(x).[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169736654969\" class=\"key-equations\">\r\n<h1>Key Equations<\/h1>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>The chain rule<\/strong>\r\n[latex]{h}^{\\prime }(x)={f}^{\\prime }(g(x)){g}^{\\prime }(x)[\/latex]<\/li>\r\n \t<li><strong>The power rule for functions<\/strong>\r\n[latex]{h}^{\\prime }(x)=n{(g(x))}^{n-1}{g}^{\\prime }(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169736655110\" class=\"textbox exercises\">\r\n<p id=\"fs-id1169736655114\">For the following exercises, given [latex]y=f(u)[\/latex] and [latex]u=g(x),[\/latex] find [latex]\\frac{dy}{dx}[\/latex] by using Leibniz\u2019s notation for the chain rule: [latex]\\frac{dy}{dx}=\\frac{dy}{du}\\frac{du}{dx}.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169736659368\" class=\"exercise\">\r\n<div id=\"fs-id1169736659370\" class=\"textbox\">\r\n<p id=\"fs-id1169736659372\">[latex]y=3u-6,u=2{x}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736659427\" class=\"exercise\">\r\n<div id=\"fs-id1169736659430\" class=\"textbox\">\r\n<p id=\"fs-id1169736659432\">[latex]y=6{u}^{3},u=7x-4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736659466\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736659466\"]\r\n<p id=\"fs-id1169736659466\">[latex]18{u}^{2}\u00b77=18{(7x-4)}^{2}\u00b77[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739281036\" class=\"exercise\">\r\n<div id=\"fs-id1169739281038\" class=\"textbox\">\r\n<p id=\"fs-id1169739281040\">[latex]y= \\sin u,u=5x-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739281115\" class=\"exercise\">\r\n<div id=\"fs-id1169739281117\" class=\"textbox\">\r\n<p id=\"fs-id1169739281119\">[latex]y= \\cos u,u=\\frac{\\text{\u2212}x}{8}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739281153\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739281153\"]\r\n<p id=\"fs-id1169739281153\">[latex]\\text{\u2212} \\sin u\u00b7\\frac{-1}{8}=\\text{\u2212} \\sin (\\frac{\\text{\u2212}x}{8})\u00b7\\frac{-1}{8}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739281207\" class=\"exercise\">\r\n<div id=\"fs-id1169739281210\" class=\"textbox\">\r\n<p id=\"fs-id1169739281212\">[latex]y= \\tan u,u=9x+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739262398\" class=\"exercise\">\r\n<div id=\"fs-id1169739262400\" class=\"textbox\">\r\n<p id=\"fs-id1169739262402\">[latex]y=\\sqrt{4u+3},u={x}^{2}-6x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739262442\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739262442\"]\r\n<p id=\"fs-id1169739262442\">[latex]\\frac{8x-24}{2\\sqrt{4u+3}}=\\frac{4x-12}{\\sqrt{4{x}^{2}-24x+3}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169739262507\">For each of the following exercises,<\/p>\r\n\r\n<ol id=\"fs-id1169739262511\" style=\"list-style-type: lower-alpha\">\r\n \t<li>decompose each function in the form [latex]y=f(u)[\/latex] and [latex]u=g(x),[\/latex] and<\/li>\r\n \t<li>find [latex]\\frac{dy}{dx}[\/latex] as a function of [latex]x.[\/latex]<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1169736602724\" class=\"exercise\">\r\n<div id=\"fs-id1169736602726\" class=\"textbox\">\r\n<p id=\"fs-id1169736602729\">[latex]y={(3x-2)}^{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736602828\" class=\"exercise\">\r\n<div id=\"fs-id1169736602830\" class=\"textbox\">\r\n<p id=\"fs-id1169736602832\">[latex]y={(3{x}^{2}+1)}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736602866\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736602866\"]\r\n<p id=\"fs-id1169736602866\">a. [latex]u=3{x}^{2}+1;[\/latex] b. [latex]18x{(3{x}^{2}+1)}^{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736602922\" class=\"exercise\">\r\n<div id=\"fs-id1169736602924\" class=\"textbox\">\r\n<p id=\"fs-id1169736588867\">[latex]y={ \\sin }^{5}(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736588943\" class=\"exercise\">\r\n<div id=\"fs-id1169736588945\" class=\"textbox\">\r\n<p id=\"fs-id1169736588947\">[latex]y={(\\frac{x}{7}+\\frac{7}{x})}^{7}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736588982\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736588982\"]\r\n<p id=\"fs-id1169736588982\">a. [latex]f(u)={u}^{7},u=\\frac{x}{7}+\\frac{7}{x};[\/latex] b. [latex]7{(\\frac{x}{7}+\\frac{7}{x})}^{6}\u00b7(\\frac{1}{7}-\\frac{7}{{x}^{2}})[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736589083\" class=\"exercise\">\r\n<div id=\"fs-id1169736589085\" class=\"textbox\">\r\n<p id=\"fs-id1169736589087\">[latex]y= \\tan ( \\sec x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739251116\" class=\"exercise\">\r\n<div id=\"fs-id1169739251118\" class=\"textbox\">\r\n<p id=\"fs-id1169739251120\">[latex]y= \\csc (\\pi x+1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739251149\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739251149\"]\r\n<p id=\"fs-id1169739251149\">a. [latex]f(u)= \\csc u,u=\\pi x+1;[\/latex] b. [latex]\\text{\u2212}\\pi \\csc (\\pi x+1)\u00b7 \\cot (\\pi x+1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739280322\" class=\"exercise\">\r\n<div id=\"fs-id1169739280325\" class=\"textbox\">\r\n<p id=\"fs-id1169739280327\">[latex]y={ \\cot }^{2}x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739280417\" class=\"exercise\">\r\n<div id=\"fs-id1169739280419\" class=\"textbox\">\r\n<p id=\"fs-id1169739280421\">[latex]y=-6{ \\sin }^{-3}x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739280445\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739280445\"]\r\n<p id=\"fs-id1169739280445\">a. [latex]f(u)=-6{u}^{-3},u= \\sin x,[\/latex] b. [latex]18{ \\sin }^{-4}x\u00b7 \\cos x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169739280513\">For the following exercises, find [latex]\\frac{dy}{dx}[\/latex] for each function.<\/p>\r\n\r\n<div id=\"fs-id1169736653150\" class=\"exercise\">\r\n<div id=\"fs-id1169736653152\" class=\"textbox\">\r\n<p id=\"fs-id1169736653155\">[latex]y={(3{x}^{2}+3x-1)}^{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736653250\" class=\"exercise\">\r\n<div id=\"fs-id1169736653252\" class=\"textbox\">\r\n<p id=\"fs-id1169736653254\">[latex]y={(5-2x)}^{-2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736653286\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736653286\"]\r\n<p id=\"fs-id1169736653286\">[latex]\\frac{4}{{(5-2x)}^{3}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736653316\" class=\"exercise\">\r\n<div id=\"fs-id1169736653318\" class=\"textbox\">\r\n<p id=\"fs-id1169736653320\">[latex]y={ \\cos }^{3}(\\pi x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739195140\" class=\"exercise\">\r\n<div id=\"fs-id1169739195142\" class=\"textbox\">\r\n<p id=\"fs-id1169739195144\">[latex]y={(2{x}^{3}-{x}^{2}+6x+1)}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739195192\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739195192\"]\r\n<p id=\"fs-id1169739195192\">[latex]6{(2{x}^{3}-{x}^{2}+6x+1)}^{2}(3{x}^{2}-x+3)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739195259\" class=\"exercise\">\r\n<div id=\"fs-id1169739195261\" class=\"textbox\">\r\n<p id=\"fs-id1169739195263\">[latex]y=\\frac{1}{{ \\sin }^{2}(x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739195337\" class=\"exercise\">\r\n<div id=\"fs-id1169739195339\" class=\"textbox\">\r\n<p id=\"fs-id1169739195341\">[latex]y={( \\tan x+ \\sin x)}^{-3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738989496\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738989496\"]\r\n<p id=\"fs-id1169738989496\">[latex]-3{( \\tan x+ \\sin x)}^{-4}\u00b7({ \\sec }^{2}x+ \\cos x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738989560\" class=\"exercise\">\r\n<div id=\"fs-id1169738989562\" class=\"textbox\">\r\n<p id=\"fs-id1169738989564\">[latex]y={x}^{2}{ \\cos }^{4}x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738989642\" class=\"exercise\">\r\n<div id=\"fs-id1169738989645\" class=\"textbox\">\r\n<p id=\"fs-id1169738989647\">[latex]y= \\sin ( \\cos 7x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738989676\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738989676\"]\r\n<p id=\"fs-id1169738989676\">[latex]-7 \\cos ( \\cos 7x)\u00b7 \\sin 7x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736585006\" class=\"exercise\">\r\n<div id=\"fs-id1169736585009\" class=\"textbox\">\r\n<p id=\"fs-id1169736585011\">[latex]y=\\sqrt{6+ \\sec \\pi {x}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736585115\" class=\"exercise\">\r\n<div id=\"fs-id1169736585117\" class=\"textbox\">\r\n<p id=\"fs-id1169736585119\">[latex]y={ \\cot }^{3}(4x+1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736585152\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736585152\"]\r\n<p id=\"fs-id1169736585152\">[latex]-12{ \\cot }^{2}(4x+1)\u00b7{ \\csc }^{2}(4x+1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738988193\" class=\"exercise\">\r\n<div id=\"fs-id1169738988195\" class=\"textbox\">\r\n<p id=\"fs-id1169738988197\">Let [latex]y={\\left[f(x)\\right]}^{3}[\/latex] and suppose that [latex]{f}^{\\prime }(1)=4[\/latex] and [latex]\\frac{dy}{dx}=10[\/latex] for [latex]x=1.[\/latex] Find [latex]f(1).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738988313\" class=\"exercise\">\r\n<div id=\"fs-id1169738988315\" class=\"textbox\">\r\n<p id=\"fs-id1169738988317\">Let [latex]y={(f(x)+5{x}^{2})}^{4}[\/latex] and suppose that [latex]f(-1)=-4[\/latex] and [latex]\\frac{dy}{dx}=3[\/latex] when [latex]x=-1.[\/latex] Find [latex]{f}^{\\prime }(-1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739374485\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739374485\"]\r\n<p id=\"fs-id1169739374485\">[latex]10\\frac{3}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739374499\" class=\"exercise\">\r\n<div id=\"fs-id1169739374501\" class=\"textbox\">\r\n<p id=\"fs-id1169739374503\">Let [latex]y={(f(u)+3x)}^{2}[\/latex] and [latex]u={x}^{3}-2x.[\/latex] If [latex]f(4)=6[\/latex] and [latex]\\frac{dy}{dx}=18[\/latex] when [latex]x=2,[\/latex] find [latex]{f}^{\\prime }(4).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739374644\" class=\"exercise\">\r\n<div id=\"fs-id1169739374646\" class=\"textbox\">\r\n<p id=\"fs-id1169739374648\"><strong>[T]<\/strong> Find the equation of the tangent line to [latex]y=\\text{\u2212} \\sin (\\frac{x}{2})[\/latex] at the origin. Use a calculator to graph the function and the tangent line together.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739374682\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739374682\"]\r\n<p id=\"fs-id1169739374682\">[latex]y=\\frac{-1}{2}x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738991878\" class=\"exercise\">\r\n<div id=\"fs-id1169738991880\" class=\"textbox\">\r\n<p id=\"fs-id1169738991882\"><strong>[T]<\/strong> Find the equation of the tangent line to [latex]y={(3x+\\frac{1}{x})}^{2}[\/latex] at the point [latex](1,16).[\/latex] Use a calculator to graph the function and the tangent line together.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738991955\" class=\"exercise\">\r\n<div id=\"fs-id1169738991958\" class=\"textbox\">\r\n<p id=\"fs-id1169738991960\">Find the [latex]x[\/latex]-coordinates at which the tangent line to [latex]y={(x-\\frac{6}{x})}^{8}[\/latex] is horizontal.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169738991998\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738991998\"]\r\n<p id=\"fs-id1169738991998\">[latex]x=\u00b1\\sqrt{6}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738992014\" class=\"exercise\">\r\n<div id=\"fs-id1169738992016\" class=\"textbox\">\r\n<p id=\"fs-id1169738992018\"><strong>[T]<\/strong> Find an equation of the line that is normal to [latex]g(\\theta )={ \\sin }^{2}(\\pi \\theta )[\/latex] at the point [latex](\\frac{1}{4},\\frac{1}{2}).[\/latex] Use a calculator to graph the function and the normal line together.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169736582610\">For the following exercises, use the information in the following table to find [latex]{h}^{\\prime }(a)[\/latex] at the given value for [latex]a.[\/latex]<\/p>\r\n\r\n<table id=\"fs-id1169736582646\" class=\"unnumbered\" summary=\"This table has five rows and five columns. The first row is a header row and it labels each column. The column headers from left to right are x, f(x), f\u2019(x), g(x), and g\u2019(x). Under the first column are the values 0, 1, 2, and 3. Under the second column are the values 2, 1, 4, and 3. Under the third column are the values 5, \u22122, 4, and \u22123. Under the fourth column are the values 0, 3, 1, and 2. Under the fifth column are g\u2019(x) are the values 2, 0, \u22121, and 3.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)[\/latex]<\/th>\r\n<th>[latex]f\\prime (x)[\/latex]<\/th>\r\n<th>[latex]g(x)[\/latex]<\/th>\r\n<th>[latex]g\\prime (x)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>0<\/td>\r\n<td>2<\/td>\r\n<td>5<\/td>\r\n<td>0<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>1<\/td>\r\n<td>1<\/td>\r\n<td>\u22122<\/td>\r\n<td>3<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>4<\/td>\r\n<td>1<\/td>\r\n<td>\u22121<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>3<\/td>\r\n<td>3<\/td>\r\n<td>\u22123<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-id1169739243334\" class=\"exercise\">\r\n<div id=\"fs-id1169739243336\" class=\"textbox\">\r\n<p id=\"fs-id1169739243338\">[latex]h(x)=f(g(x));a=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739243384\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739243384\"]\r\n<p id=\"fs-id1169739243384\">10<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739243389\" class=\"exercise\">\r\n<div id=\"fs-id1169739243391\" class=\"textbox\">\r\n<p id=\"fs-id1169739243393\">[latex]h(x)=g(f(x));a=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736662054\" class=\"exercise\">\r\n<div id=\"fs-id1169736662056\" class=\"textbox\">\r\n<p id=\"fs-id1169736662058\">[latex]h(x)={({x}^{4}+g(x))}^{-2};a=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736662118\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736662118\"]\r\n<p id=\"fs-id1169736662118\">[latex]-\\frac{1}{8}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736662132\" class=\"exercise\">\r\n<div id=\"fs-id1169736662134\" class=\"textbox\">\r\n<p id=\"fs-id1169736662136\">[latex]h(x)={(\\frac{f(x)}{g(x)})}^{2};a=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736662220\" class=\"exercise\">\r\n<div id=\"fs-id1169736662223\" class=\"textbox\">\r\n<p id=\"fs-id1169736662225\">[latex]h(x)=f(x+f(x));a=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736608228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736608228\"]\r\n<p id=\"fs-id1169736608228\">-4<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736608237\" class=\"exercise\">\r\n<div id=\"fs-id1169736608239\" class=\"textbox\">\r\n<p id=\"fs-id1169736608241\">[latex]h(x)={(1+g(x))}^{3};a=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736608305\" class=\"exercise\">\r\n<div id=\"fs-id1169736608307\" class=\"textbox\">\r\n<p id=\"fs-id1169736608309\">[latex]h(x)=g(2+f({x}^{2}));a=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736608364\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736608364\"]\r\n<p id=\"fs-id1169736608364\">-12<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736608372\" class=\"exercise\">\r\n<div id=\"fs-id1169736608374\" class=\"textbox\">\r\n<p id=\"fs-id1169736608376\">[latex]h(x)=f(g( \\sin x));a=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736608434\" class=\"exercise\">\r\n<div id=\"fs-id1169736592171\" class=\"textbox\">\r\n<p id=\"fs-id1169736592173\"><strong>[T]<\/strong> The position function of a freight train is given by [latex]s(t)=100{(t+1)}^{-2},[\/latex] with [latex]s[\/latex] in meters and [latex]t[\/latex] in seconds. At time [latex]t=6[\/latex] s, find the train\u2019s<\/p>\r\n\r\n<ol id=\"fs-id1169736592236\" style=\"list-style-type: lower-alpha\">\r\n \t<li>velocity and<\/li>\r\n \t<li>acceleration.<\/li>\r\n \t<li>Using a. and b. is the train speeding up or slowing down?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736592256\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736592256\"]\r\n<p id=\"fs-id1169736592256\">a. [latex]-\\frac{200}{343}[\/latex] m\/s, b. [latex]\\frac{600}{2401}[\/latex] m\/s<sup>2<\/sup>, c. The train is slowing down since velocity and acceleration have opposite signs.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736592288\" class=\"exercise\">\r\n<div id=\"fs-id1169736592290\" class=\"textbox\">\r\n<p id=\"fs-id1169736592292\"><strong>[T]<\/strong> A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where [latex]t[\/latex] is measured in seconds and [latex]s[\/latex] is in inches:<\/p>\r\n<p id=\"fs-id1169736592309\">[latex]s(t)=-3 \\cos (\\pi t+\\frac{\\pi }{4}).[\/latex]<\/p>\r\n\r\n<ol id=\"fs-id1169736592353\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Determine the position of the spring at [latex]t=1.5[\/latex] s.<\/li>\r\n \t<li>Find the velocity of the spring at [latex]t=1.5[\/latex] s.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736592402\" class=\"exercise\">\r\n<div id=\"fs-id1169736616212\" class=\"textbox\">\r\n<p id=\"fs-id1169736616214\"><strong>[T]<\/strong> The total cost to produce [latex]x[\/latex] boxes of Thin Mint Girl Scout cookies is [latex]C[\/latex] dollars, where [latex]C=0.0001{x}^{3}-0.02{x}^{2}+3x+300.[\/latex] In [latex]t[\/latex] weeks production is estimated to be [latex]x=1600+100t[\/latex] boxes.<\/p>\r\n\r\n<ol id=\"fs-id1169736616286\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find the marginal cost [latex]{C}^{\\prime }(x).[\/latex]<\/li>\r\n \t<li>Use Leibniz\u2019s notation for the chain rule, [latex]\\frac{dC}{dt}=\\frac{dC}{dx}\u00b7\\frac{dx}{dt},[\/latex] to find the rate with respect to time [latex]t[\/latex] that the cost is changing.<\/li>\r\n \t<li>Use b. to determine how fast costs are increasing when [latex]t=2[\/latex] weeks. Include units with the answer.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736616383\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736616383\"]\r\n<p id=\"fs-id1169736616383\">a. [latex]{C}^{\\prime }(x)=0.0003{x}^{2}-0.04x+3[\/latex] b. [latex]\\frac{dC}{dt}=100\u00b7(0.0003{x}^{2}-0.04x+3)[\/latex] c. Approximately $90,300 per week<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736618622\" class=\"exercise\">\r\n<div id=\"fs-id1169736618624\" class=\"textbox\">\r\n<p id=\"fs-id1169736618626\"><strong>[T]<\/strong> The formula for the area of a circle is [latex]A=\\pi {r}^{2},[\/latex] where [latex]r[\/latex] is the radius of the circle. Suppose a circle is expanding, meaning that both the area [latex]A[\/latex] and the radius [latex]r[\/latex] (in inches) are expanding.<\/p>\r\n\r\n<ol id=\"fs-id1169736618664\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Suppose [latex]r=2-\\frac{100}{{(t+7)}^{2}}[\/latex] where [latex]t[\/latex] is time in seconds. Use the chain rule [latex]\\frac{dA}{dt}=\\frac{dA}{dr}\u00b7\\frac{dr}{dt}[\/latex] to find the rate at which the area is expanding.<\/li>\r\n \t<li>Use a. to find the rate at which the area is expanding at [latex]t=4[\/latex] s.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736618829\" class=\"exercise\">\r\n<div id=\"fs-id1169736618831\" class=\"textbox\">\r\n<p id=\"fs-id1169736618834\"><strong>[T]<\/strong> The formula for the volume of a sphere is [latex]S=\\frac{4}{3}\\pi {r}^{3},[\/latex] where [latex]r[\/latex] (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.<\/p>\r\n\r\n<ol id=\"fs-id1169739066678\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Suppose [latex]r=\\frac{1}{{(t+1)}^{2}}-\\frac{1}{12}[\/latex] where [latex]t[\/latex] is time in minutes. Use the chain rule [latex]\\frac{dS}{dt}=\\frac{dS}{dr}\u00b7\\frac{dr}{dt}[\/latex] to find the rate at which the snowball is melting.<\/li>\r\n \t<li>Use a. to find the rate at which the volume is changing at [latex]t=1[\/latex] min.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739066789\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739066789\"]\r\n<p id=\"fs-id1169739066789\">a. [latex]\\frac{dS}{dt}=-\\frac{8\\pi {r}^{2}}{{(t+1)}^{3}}[\/latex] b. The volume is decreasing at a rate of [latex]-\\frac{\\pi }{36}[\/latex] ft<sup>3<\/sup>\/min.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739066858\" class=\"exercise\">\r\n<div id=\"fs-id1169739066860\" class=\"textbox\">\r\n<p id=\"fs-id1169739066862\"><strong>[T]<\/strong> The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function [latex]T(x)=94-10 \\cos \\left[\\frac{\\pi }{12}(x-2)\\right],[\/latex] where [latex]x[\/latex] is hours after midnight. Find the rate at which the temperature is changing at 4 p.m.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739296655\" class=\"exercise\">\r\n<div id=\"fs-id1169739296657\" class=\"textbox\">\r\n<p id=\"fs-id1169739296659\"><strong>[T]<\/strong> The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function [latex]D(t)=5 \\sin (\\frac{\\pi }{6}t-\\frac{7\\pi }{6})+8,[\/latex] where [latex]t[\/latex] is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739296729\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739296729\"]\r\n<p id=\"fs-id1169739296729\">[latex]~2.3[\/latex] ft\/hr<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169739296745\" class=\"definition\">\r\n \t<dt>chain rule<\/dt>\r\n \t<dd id=\"fs-id1169739296750\">the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>State the chain rule for the composition of two functions.<\/li>\n<li>Apply the chain rule together with the power rule.<\/li>\n<li>Apply the chain rule and the product\/quotient rules correctly in combination when both are necessary.<\/li>\n<li>Recognize the chain rule for a composition of three or more functions.<\/li>\n<li>Describe the proof of the chain rule.<\/li>\n<\/ul>\n<\/div>\n<p>We have seen the techniques for differentiating basic functions [latex]({x}^{n}, \\sin x, \\cos x,\\text{etc}.)[\/latex] as well as sums, differences, products, quotients, and constant multiples of these functions. However, these techniques do not allow us to differentiate compositions of functions, such as [latex]h(x)= \\sin ({x}^{3})[\/latex] or [latex]k(x)=\\sqrt{3{x}^{2}+1}.[\/latex] In this section, we study the rule for finding the derivative of the composition of two or more functions.<\/p>\n<div id=\"fs-id1169738923754\" class=\"bc-section section\">\n<h1>Deriving the Chain Rule<\/h1>\n<p id=\"fs-id1169738824611\">When we have a function that is a composition of two or more functions, we could use all of the techniques we have already learned to differentiate it. However, using all of those techniques to break down a function into simpler parts that we are able to differentiate can get cumbersome. Instead, we use the <strong>chain rule<\/strong>, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.<\/p>\n<p id=\"fs-id1169736656600\">To put this rule into context, let\u2019s take a look at an example: [latex]h(x)= \\sin ({x}^{3}).[\/latex] We can think of the derivative of this function with respect to [latex]x[\/latex] as the rate of change of [latex]\\sin ({x}^{3})[\/latex] relative to the change in [latex]x.[\/latex] Consequently, we want to know how [latex]\\sin ({x}^{3})[\/latex] changes as [latex]x[\/latex] changes. We can think of this event as a chain reaction: As [latex]x[\/latex] changes, [latex]{x}^{3}[\/latex] changes, which leads to a change in [latex]\\sin ({x}^{3}).[\/latex] This chain reaction gives us hints as to what is involved in computing the derivative of [latex]\\sin ({x}^{3}).[\/latex] First of all, a change in [latex]x[\/latex] forcing a change in [latex]{x}^{3}[\/latex] suggests that somehow the derivative of [latex]{x}^{3}[\/latex] is involved. In addition, the change in [latex]{x}^{3}[\/latex] forcing a change in [latex]\\sin ({x}^{3})[\/latex] suggests that the derivative of [latex]\\sin (u)[\/latex] with respect to [latex]u,[\/latex] where [latex]u={x}^{3},[\/latex] is also part of the final derivative.<\/p>\n<p id=\"fs-id1169738823074\">We can take a more formal look at the derivative of [latex]h(x)= \\sin ({x}^{3})[\/latex] by setting up the limit that would give us the derivative at a specific value [latex]a[\/latex] in the domain of [latex]h(x)= \\sin ({x}^{3}).[\/latex]<\/p>\n<div id=\"fs-id1169738941291\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(a)=\\underset{x\\to a}{\\text{lim}}\\frac{ \\sin ({x}^{3})- \\sin ({a}^{3})}{x-a}.[\/latex]<\/div>\n<p id=\"fs-id1169738954231\">This expression does not seem particularly helpful; however, we can modify it by multiplying and dividing by the expression [latex]{x}^{3}-{a}^{3}[\/latex] to obtain<\/p>\n<div id=\"fs-id1169739101346\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(a)=\\underset{x\\to a}{\\text{lim}}\\frac{ \\sin ({x}^{3})- \\sin ({a}^{3})}{{x}^{3}-{a}^{3}}\u00b7\\frac{{x}^{3}-{a}^{3}}{x-a}.[\/latex]<\/div>\n<p id=\"fs-id1169738969849\">From the definition of the derivative, we can see that the second factor is the derivative of [latex]{x}^{3}[\/latex] at [latex]x=a.[\/latex] That is,<\/p>\n<div id=\"fs-id1169739096283\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\text{lim}}\\frac{{x}^{3}-{a}^{3}}{x-a}=\\frac{d}{dx}({x}^{3})=3{a}^{2}.[\/latex]<\/div>\n<p id=\"fs-id1169738971694\">However, it might be a little more challenging to recognize that the first term is also a derivative. We can see this by letting [latex]u={x}^{3}[\/latex] and observing that as [latex]x\\to a,u\\to {a}^{3}\\text{:}[\/latex]<\/p>\n<div id=\"fs-id1169739035291\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\underset{x\\to a}{\\text{lim}}\\frac{ \\sin ({x}^{3})- \\sin ({a}^{3})}{{x}^{3}-{a}^{3}}& =\\underset{u\\to {a}^{3}}{\\text{lim}}\\frac{ \\sin u- \\sin ({a}^{3})}{u-{a}^{3}}\\hfill \\\\ & =\\frac{d}{du}{( \\sin u)}_{u={a}^{3}}\\hfill \\\\ & = \\cos ({a}^{3}).\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1169739000166\">Thus, [latex]{h}^{\\prime }(a)= \\cos ({a}^{3})\u00b73{a}^{2}.[\/latex]<\/p>\n<p id=\"fs-id1169739044550\">In other words, if [latex]h(x)= \\sin ({x}^{3}),[\/latex] then [latex]{h}^{\\prime }(x)= \\cos ({x}^{3})\u00b73{x}^{2}.[\/latex] Thus, if we think of [latex]h(x)= \\sin ({x}^{3})[\/latex] as the composition [latex](f\\circ g)(x)=f(g(x))[\/latex] where [latex]f(x)=[\/latex] sin [latex]x[\/latex] and [latex]g(x)={x}^{3},[\/latex] then the derivative of [latex]h(x)= \\sin ({x}^{3})[\/latex] is the product of the derivative of [latex]g(x)={x}^{3}[\/latex] and the derivative of the function [latex]f(x)= \\sin x[\/latex] evaluated at the function [latex]g(x)={x}^{3}.[\/latex] At this point, we anticipate that for [latex]h(x)= \\sin (g(x)),[\/latex] it is quite likely that [latex]{h}^{\\prime }(x)= \\cos (g(x)){g}^{\\prime }(x).[\/latex] As we determined above, this is the case for [latex]h(x)= \\sin ({x}^{3}).[\/latex]<\/p>\n<p id=\"fs-id1169738955406\">Now that we have derived a special case of the chain rule, we state the general case and then apply it in a general form to other composite functions. An informal proof is provided at the end of the section.<\/p>\n<div id=\"fs-id1169739194586\" class=\"textbox key-takeaways\">\n<h3>Rule: The Chain Rule<\/h3>\n<p id=\"fs-id1169736612521\">Let [latex]f[\/latex] and [latex]g[\/latex] be functions. For all [latex]x[\/latex] in the domain of [latex]g[\/latex] for which [latex]g[\/latex] is differentiable at [latex]x[\/latex] and [latex]f[\/latex] is differentiable at [latex]g(x),[\/latex] the derivative of the composite function<\/p>\n<div id=\"fs-id1169739233799\" class=\"equation unnumbered\">[latex]h(x)=(f\\circ g)(x)=f(g(x))[\/latex]<\/div>\n<p id=\"fs-id1169738949353\">is given by<\/p>\n<div id=\"fs-id1169738948938\" class=\"equation\">[latex]{h}^{\\prime }(x)={f}^{\\prime }(g(x)){g}^{\\prime }(x).[\/latex]<\/div>\n<p id=\"fs-id1169739199751\">Alternatively, if [latex]y[\/latex] is a function of [latex]u,[\/latex] and [latex]u[\/latex] is a function of [latex]x,[\/latex] then<\/p>\n<div id=\"fs-id1169739187558\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=\\frac{dy}{du}\u00b7\\frac{du}{dx}.[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1169738971083\" class=\"textbox tryit media-2\">\n<p id=\"fs-id1169739080685\">Watch an <a href=\"http:\/\/www.openstaxcollege.org\/l\/20_chainrule2\">animation<\/a> of the chain rule.<\/p>\n<\/div>\n<div id=\"fs-id1169736619728\" class=\"textbox key-takeaways problem-solving\">\n<h3>Problem-Solving Strategy: Applying the Chain Rule<\/h3>\n<ol id=\"fs-id1169739182671\">\n<li>To differentiate [latex]h(x)=f(g(x)),[\/latex] begin by identifying [latex]f(x)[\/latex] and [latex]g(x).[\/latex]<\/li>\n<li>Find [latex]f\\prime (x)[\/latex] and evaluate it at [latex]g(x)[\/latex] to obtain [latex]{f}^{\\prime }(g(x)).[\/latex]<\/li>\n<li>Find [latex]{g}^{\\prime }(x).[\/latex]<\/li>\n<li>Write [latex]{h}^{\\prime }(x)={f}^{\\prime }(g(x))\u00b7{g}^{\\prime }(x).[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1169739269865\"><em>Note<\/em>: When applying the chain rule to the composition of two or more functions, keep in mind that we work our way from the outside function in. It is also useful to remember that the derivative of the composition of two functions can be thought of as having two parts; the derivative of the composition of three functions has three parts; and so on. Also, remember that we never evaluate a derivative at a derivative.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739001005\" class=\"bc-section section\">\n<h1>The Chain and Power Rules Combined<\/h1>\n<p id=\"fs-id1169739096228\">We can now apply the chain rule to composite functions, but note that we often need to use it with other rules. For example, to find derivatives of functions of the form [latex]h(x)={(g(x))}^{n},[\/latex] we need to use the chain rule combined with the power rule. To do so, we can think of [latex]h(x)={(g(x))}^{n}[\/latex] as [latex]f(g(x))[\/latex] where [latex]f(x)={x}^{n}.[\/latex] Then [latex]{f}^{\\prime }(x)=n{x}^{n-1}.[\/latex] Thus, [latex]{f}^{\\prime }(g(x))=n{(g(x))}^{n-1}.[\/latex] This leads us to the derivative of a power function using the chain rule,<\/p>\n<div id=\"fs-id1169739325710\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)=n{(g(x))}^{n-1}{g}^{\\prime }(x)[\/latex]<\/div>\n<div id=\"fs-id1169739187734\" class=\"textbox key-takeaways\">\n<h3>Rule: Power Rule for Composition of Functions<\/h3>\n<p id=\"fs-id1169738978364\">For all values of [latex]x[\/latex] for which the derivative is defined, if<\/p>\n<div id=\"fs-id1169739006308\" class=\"equation unnumbered\">[latex]h(x)={(g(x))}^{n}.[\/latex]<\/div>\n<p id=\"fs-id1169739242349\">Then<\/p>\n<div id=\"fs-id1169739222795\" class=\"equation\">[latex]{h}^{\\prime }(x)=n{(g(x))}^{n-1}{g}^{\\prime }(x).[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1169739274312\" class=\"textbox examples\">\n<h3>Using the Chain and Power Rules<\/h3>\n<div id=\"fs-id1169739274625\" class=\"exercise\">\n<div id=\"fs-id1169739282636\" class=\"textbox\">\n<p id=\"fs-id1169736589119\">Find the derivative of [latex]h(x)=\\frac{1}{{(3{x}^{2}+1)}^{2}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736658840\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736658840\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736658840\">First, rewrite [latex]h(x)=\\frac{1}{{(3{x}^{2}+1)}^{2}}={(3{x}^{2}+1)}^{-2}.[\/latex]<\/p>\n<p id=\"fs-id1169739333152\">Applying the power rule with [latex]g(x)=3{x}^{2}+1,[\/latex] we have<\/p>\n<div id=\"fs-id1169736609881\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)=-2{(3{x}^{2}+1)}^{-3}(6x).[\/latex]<\/div>\n<p id=\"fs-id1169736655793\">Rewriting back to the original form gives us<\/p>\n<div id=\"fs-id1169739008104\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)=\\frac{-12x}{{(3{x}^{2}+1)}^{3}}.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736662938\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169736662942\" class=\"exercise\">\n<div id=\"fs-id1169736595959\" class=\"textbox\">\n<p id=\"fs-id1169736595961\">Find the derivative of [latex]h(x)={(2{x}^{3}+2x-1)}^{4}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739325717\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739325717\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739325717\">[latex]{h}^{\\prime }(x)=4{(2{x}^{3}+2x-1)}^{3}(6x+2)=8(3x+1){(2{x}^{3}+2x-1)}^{3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169739274677\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169739179049\">Use <a class=\"autogenerated-content\" href=\"#fs-id1169739222795\">(Figure)<\/a> with [latex]g(x)=2{x}^{3}+2x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739302258\" class=\"textbox examples\">\n<h3>Using the Chain and Power Rules with a Trigonometric Function<\/h3>\n<div id=\"fs-id1169736609872\" class=\"exercise\">\n<div id=\"fs-id1169736609874\" class=\"textbox\">\n<p id=\"fs-id1169739273001\">Find the derivative of [latex]h(x)={ \\sin }^{3}x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739182335\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739182335\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739182335\">First recall that [latex]{ \\sin }^{3}x={( \\sin x)}^{3},[\/latex] so we can rewrite [latex]h(x)={ \\sin }^{3}x[\/latex] as [latex]h(x)={( \\sin x)}^{3}.[\/latex]<\/p>\n<p id=\"fs-id1169739351647\">Applying the power rule with [latex]g(x)= \\sin x,[\/latex] we obtain<\/p>\n<div id=\"fs-id1169739039132\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)=3{( \\sin x)}^{2} \\cos x=3{ \\sin }^{2}x \\cos x.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739190027\" class=\"textbox examples\">\n<h3>Finding the Equation of a Tangent Line<\/h3>\n<div id=\"fs-id1169739190029\" class=\"exercise\">\n<div id=\"fs-id1169736613820\" class=\"textbox\">\n<p id=\"fs-id1169736613825\">Find the equation of a line tangent to the graph of [latex]h(x)=\\frac{1}{{(3x-5)}^{2}}[\/latex] at [latex]x=2.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739336066\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739336066\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739336066\">Because we are finding an equation of a line, we need a point. The [latex]x[\/latex]-coordinate of the point is 2. To find the [latex]y[\/latex]-coordinate, substitute 2 into [latex]h(x).[\/latex] Since [latex]h(2)=\\frac{1}{{(3(2)-5)}^{2}}=1,[\/latex] the point is [latex](2,1).[\/latex]<\/p>\n<p id=\"fs-id1169736607681\">For the slope, we need [latex]{h}^{\\prime }(2).[\/latex] To find [latex]{h}^{\\prime }(x),[\/latex] first we rewrite [latex]h(x)={(3x-5)}^{-2}[\/latex] and apply the power rule to obtain<\/p>\n<div id=\"fs-id1169736662505\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)=-2{(3x-5)}^{-3}(3)=-6{(3x-5)}^{-3}.[\/latex]<\/div>\n<p id=\"fs-id1169739111347\">By substituting, we have [latex]{h}^{\\prime }(2)=-6{(3(2)-5)}^{-3}=-6.[\/latex] Therefore, the line has equation [latex]y-1=-6(x-2).[\/latex] Rewriting, the equation of the line is [latex]y=-6x+13.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739301868\" class=\"textbox exercises checkpoint\">\n<div class=\"exercise\">\n<div id=\"fs-id1169739301873\" class=\"textbox\">\n<p id=\"fs-id1169739303885\">Find the equation of the line tangent to the graph of [latex]f(x)={({x}^{2}-2)}^{3}[\/latex] at [latex]x=-2.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738869705\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738869705\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738869705\">[latex]y=-48x-88[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169739242781\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169739252012\">Use the preceding example as a guide.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739252019\" class=\"bc-section section\">\n<h1>Combining the Chain Rule with Other Rules<\/h1>\n<p id=\"fs-id1169739293763\">Now that we can combine the chain rule and the power rule, we examine how to combine the chain rule with the other rules we have learned. In particular, we can use it with the formulas for the derivatives of trigonometric functions or with the product rule.<\/p>\n<div id=\"fs-id1169736656617\" class=\"textbox examples\">\n<h3>Using the Chain Rule on a General Cosine Function<\/h3>\n<div id=\"fs-id1169736656619\" class=\"exercise\">\n<div id=\"fs-id1169736656621\" class=\"textbox\">\n<p id=\"fs-id1169739327889\">Find the derivative of [latex]h(x)= \\cos (g(x)).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169736657088\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736657088\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736657088\" class=\"hidden-answer\" style=\"display: none\">Think of [latex]h(x)= \\cos (g(x))[\/latex] as [latex]f(g(x))[\/latex] where [latex]f(x)= \\cos x.[\/latex] Since [latex]{f}^{\\prime }(x)=\\text{\u2212} \\sin x.[\/latex] we have [latex]{f}^{\\prime }(g(x))=\\text{\u2212} \\sin (g(x)).[\/latex] Then we do the following calculation.<\/p>\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill {h}^{\\prime }(x)& ={f}^{\\prime }(g(x)){g}^{\\prime }(x)\\hfill & & & \\text{Apply the chain rule.}\\hfill \\\\ & =\\text{\u2212} \\sin (g(x)){g}^{\\prime }(x)\\hfill & & & \\text{Substitute}{f}^{\\prime }(g(x))=\\text{\u2212} \\sin (g(x)).\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1169739285002\">Thus, the derivative of [latex]h(x)= \\cos (g(x))[\/latex] is given by [latex]{h}^{\\prime }(x)=\\text{\u2212} \\sin (g(x)){g}^{\\prime }(x).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169739301534\">In the following example we apply the rule that we have just derived.<\/p>\n<div id=\"fs-id1169739301537\" class=\"textbox examples\">\n<h3>Using the Chain Rule on a Cosine Function<\/h3>\n<div id=\"fs-id1169739301539\" class=\"exercise\">\n<div id=\"fs-id1169739301541\" class=\"textbox\">\n<p id=\"fs-id1169739301547\">Find the derivative of [latex]h(x)= \\cos (5{x}^{2}).[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>Let [latex]g(x)=5{x}^{2}.[\/latex] Then [latex]{g}^{\\prime }(x)=10x.[\/latex] Using the result from the previous example,<\/p>\n<div id=\"fs-id1169739111373\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill {h}^{\\prime }(x)& =\\text{\u2212} \\sin (5{x}^{2})\u00b710x\\hfill \\\\ & =-10x \\sin (5{x}^{2}).\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739333921\" class=\"textbox examples\">\n<h3>Using the Chain Rule on Another Trigonometric Function<\/h3>\n<div id=\"fs-id1169739333923\" class=\"exercise\">\n<div id=\"fs-id1169739333926\" class=\"textbox\">\n<p id=\"fs-id1169739333931\">Find the derivative of [latex]h(x)= \\sec (4{x}^{5}+2x).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739300092\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739300092\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739300092\">Apply the chain rule to [latex]h(x)= \\sec (g(x))[\/latex] to obtain<\/p>\n<div id=\"fs-id1169739285070\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)= \\sec (g(x) \\tan (g(x)){g}^{\\prime }(x).[\/latex]<\/div>\n<p id=\"fs-id1169736615162\">In this problem, [latex]g(x)=4{x}^{5}+2x,[\/latex] so we have [latex]{g}^{\\prime }(x)=20{x}^{4}+2.[\/latex] Therefore, we obtain<\/p>\n<div id=\"fs-id1169739299798\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill {h}^{\\prime }(x)& = \\sec (4{x}^{5}+2x) \\tan (4{x}^{5}+2x)(20{x}^{4}+2)\\hfill \\\\ & =(20{x}^{4}+2) \\sec (4{x}^{5}+2x) \\tan (4{x}^{5}+2x).\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739188144\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169739188147\" class=\"exercise\">\n<div id=\"fs-id1169739188149\" class=\"textbox\">\n<p id=\"fs-id1169739188151\">Find the derivative of [latex]h(x)= \\sin (7x+2).[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>[latex]{h}^{\\prime }(x)=7 \\cos (7x+2)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169736607572\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169736607578\">Apply the chain rule to [latex]h(x)= \\sin g(x)[\/latex] first and then use [latex]g(x)=7x+2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169736610159\">At this point we provide a list of derivative formulas that may be obtained by applying the chain rule in conjunction with the formulas for derivatives of trigonometric functions. Their derivations are similar to those used in <a class=\"autogenerated-content\" href=\"#fs-id1169736656617\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1169739333921\">(Figure)<\/a>. For convenience, formulas are also given in Leibniz\u2019s notation, which some students find easier to remember. (We discuss the chain rule using Leibniz\u2019s notation at the end of this section.) It is not absolutely necessary to memorize these as separate formulas as they are all applications of the chain rule to previously learned formulas.<\/p>\n<div id=\"fs-id1169736610174\" class=\"textbox key-takeaways theorem\">\n<h3>Using the Chain Rule with Trigonometric Functions<\/h3>\n<p id=\"fs-id1169736655841\">For all values of [latex]x[\/latex] for which the derivative is defined,<\/p>\n<div id=\"fs-id1169736655849\" class=\"equation unnumbered\">[latex]\\begin{array}{cccc}\\frac{d}{dx}( \\sin (g(x))= \\cos (g(x))g\\prime (x)\\hfill & & & \\frac{d}{dx} \\sin u= \\cos u\\frac{du}{dx}\\hfill \\\\ \\frac{d}{dx}( \\cos (g(x))=\\text{\u2212} \\sin (g(x))g\\prime (x)\\hfill & & & \\frac{d}{dx} \\cos u=\\text{\u2212} \\sin u\\frac{du}{dx}\\hfill \\\\ \\frac{d}{dx}( \\tan (g(x))\\phantom{\\rule{0.22em}{0ex}}={ \\sec }^{2}(g(x))g\\prime (x)\\hfill & & & \\frac{d}{dx} \\tan u={ \\sec }^{2}u\\frac{du}{dx}\\hfill \\\\ \\frac{d}{dx}( \\cot (g(x))\\phantom{\\rule{0.25em}{0ex}}=\\text{\u2212}{ \\csc }^{2}(g(x))g\\prime (x)\\hfill & & & \\frac{d}{dx} \\cot u\\phantom{\\rule{0.25em}{0ex}}=\\text{\u2212}{ \\csc }^{2}u\\frac{du}{dx}\\hfill \\\\ \\frac{d}{dx}( \\sec (g(x))\\phantom{\\rule{0.25em}{0ex}}= \\sec (g(x) \\tan (g(x))g\\prime (x)\\hfill & & & \\frac{d}{dx} \\sec u\\phantom{\\rule{0.25em}{0ex}}= \\sec u \\tan u\\frac{du}{dx}\\hfill \\\\ \\frac{d}{dx}( \\csc (g(x))\\phantom{\\rule{0.25em}{0ex}}=\\text{\u2212} \\csc (g(x)) \\cot (g(x))g\\prime (x)\\hfill & & & \\frac{d}{dx} \\csc u\\phantom{\\rule{0.25em}{0ex}}=\\text{\u2212} \\csc u \\cot u\\frac{du}{dx}.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1169739298047\" class=\"textbox examples\">\n<div id=\"fs-id1169739298049\" class=\"exercise\">\n<div class=\"textbox\">\n<h3>Combining the Chain Rule with the Product Rule<\/h3>\n<p id=\"fs-id1169739298056\">Find the derivative of [latex]h(x)={(2x+1)}^{5}{(3x-2)}^{7}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739345782\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739345782\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739345782\">First apply the product rule, then apply the chain rule to each term of the product.<\/p>\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill {h}^{\\prime }(x)& =\\frac{d}{dx}({(2x+1)}^{5})\u00b7{(3x-2)}^{7}+\\frac{d}{dx}({(3x-2)}^{7})\u00b7{(2x+1)}^{5}\\hfill & & & \\text{Apply the product rule.}\\hfill \\\\ & =5{(2x+1)}^{4}\u00b72\u00b7{(3x-2)}^{7}+7{(3x-2)}^{6}\u00b73\u00b7{(2x+1)}^{5}\\hfill & & & \\text{Apply the chain rule.}\\hfill \\\\ & =10{(2x+1)}^{4}{(3x-2)}^{7}+21{(3x-2)}^{6}{(2x+1)}^{5}\\hfill & & & \\text{Simplify.}\\hfill \\\\ & ={(2x+1)}^{4}{(3x-2)}^{6}(10(3x-7)+21(2x+1))\\hfill & & & \\text{Factor out}{(2x+1)}^{4}{(3x-2)}^{6}.\\hfill \\\\ & ={(2x+1)}^{4}{(3x-2)}^{6}(72x-49)\\hfill & & & \\text{Simplify.}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736603462\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169736603465\" class=\"exercise\">\n<div id=\"fs-id1169736603467\" class=\"textbox\">\n<p id=\"fs-id1169736603469\">Find the derivative of [latex]h(x)=\\frac{x}{{(2x+3)}^{3}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736603515\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736603515\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736603515\">[latex]{h}^{\\prime }(x)=\\frac{3-4x}{{(2x+3)}^{4}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169736603566\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169736603573\">Start out by applying the quotient rule. Remember to use the chain rule to differentiate the denominator.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736603581\" class=\"bc-section section\">\n<h1>Composites of Three or More Functions<\/h1>\n<p id=\"fs-id1169739304867\">We can now combine the chain rule with other rules for differentiating functions, but when we are differentiating the composition of three or more functions, we need to apply the chain rule more than once. If we look at this situation in general terms, we can generate a formula, but we do not need to remember it, as we can simply apply the chain rule multiple times.<\/p>\n<p id=\"fs-id1169739304874\">In general terms, first we let<\/p>\n<div id=\"fs-id1169739304877\" class=\"equation unnumbered\">[latex]k(x)=h(f(g(x))).[\/latex]<\/div>\n<p>Then, applying the chain rule once we obtain<\/p>\n<div id=\"fs-id1169739304927\" class=\"equation unnumbered\">[latex]{k}^{\\prime }(x)=\\frac{d}{dx}(h(f(g(x)))=h\\prime (f(g(x)))\u00b7\\frac{d}{dx}f((g(x))).[\/latex]<\/div>\n<p id=\"fs-id1169736597613\">Applying the chain rule again, we obtain<\/p>\n<div id=\"fs-id1169736597616\" class=\"equation unnumbered\">[latex]{k}^{\\prime }(x)={h}^{\\prime }(f(g(x)){f}^{\\prime }(g(x)){g}^{\\prime }(x)).[\/latex]<\/div>\n<div id=\"fs-id1169736597703\" class=\"textbox key-takeaways\">\n<h3>Rule: Chain Rule for a Composition of Three Functions<\/h3>\n<p id=\"fs-id1169736597709\">For all values of [latex]x[\/latex] for which the function is differentiable, if<\/p>\n<div id=\"fs-id1169736597716\" class=\"equation unnumbered\">[latex]k(x)=h(f(g(x))),[\/latex]<\/div>\n<p id=\"fs-id1169736597763\">then<\/p>\n<div id=\"fs-id1169736597766\" class=\"equation unnumbered\">[latex]{k}^{\\prime }(x)={h}^{\\prime }(f(g(x))){f}^{\\prime }(g(x)){g}^{\\prime }(x).[\/latex]<\/div>\n<p id=\"fs-id1169736658568\">In other words, we are applying the chain rule twice.<\/p>\n<\/div>\n<p id=\"fs-id1169736658572\">Notice that the derivative of the composition of three functions has three parts. (Similarly, the derivative of the composition of four functions has four parts, and so on.) Also, <em>remember, we can always work from the outside in, taking one derivative at a time.<\/em><\/p>\n<div id=\"fs-id1169736658580\" class=\"textbox examples\">\n<h3>Differentiating a Composite of Three Functions<\/h3>\n<div id=\"fs-id1169736658582\" class=\"exercise\">\n<div id=\"fs-id1169736658584\" class=\"textbox\">\n<p id=\"fs-id1169736658589\">Find the derivative of [latex]k(x)={ \\cos }^{4}({7x}^{2}+1).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736658640\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736658640\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736658640\">First, rewrite [latex]k(x)[\/latex] as<\/p>\n<div id=\"fs-id1169736658656\" class=\"equation unnumbered\">[latex]k(x)={( \\cos (7{x}^{2}+1))}^{4}.[\/latex]<\/div>\n<p id=\"fs-id1169739275169\">Then apply the chain rule several times.<\/p>\n<div id=\"fs-id1169739275173\" class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill {k}^{\\prime }(x)& =4{( \\cos (7{x}^{2}+1))}^{3}(\\frac{d}{dx}( \\cos (7{x}^{2}+1))\\hfill & & & \\text{Apply the chain rule.}\\hfill \\\\ & =4{( \\cos (7{x}^{2}+1))}^{3}(\\text{\u2212} \\sin (7{x}^{2}+1))(\\frac{d}{dx}(7{x}^{2}+1))\\hfill & & & \\text{Apply the chain rule.}\\hfill \\\\ & =4{( \\cos (7{x}^{2}+1))}^{3}(\\text{\u2212} \\sin (7{x}^{2}+1))(14x)\\hfill & & & \\text{Apply the chain rule.}\\hfill \\\\ & =-56x \\sin (7{x}^{2}+1){ \\cos }^{3}(7{x}^{2}+1)\\hfill & & & \\text{Simplify.}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739350744\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169739350747\" class=\"exercise\">\n<div id=\"fs-id1169739350749\" class=\"textbox\">\n<p id=\"fs-id1169739350751\">Find the derivative of [latex]h(x)={ \\sin }^{6}({x}^{3}).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739350792\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739350792\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739350792\">[latex]{h}^{\\prime }(x)=18{x}^{2}{ \\sin }^{5}({x}^{3}) \\cos ({x}^{3})[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169739350851\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169739350858\">Rewrite [latex]h(x)={ \\sin }^{6}({x}^{3})={( \\sin ({x}^{3}))}^{6}[\/latex] and use <a class=\"autogenerated-content\" href=\"#fs-id1169736658580\">(Figure)<\/a> as a guide.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736593542\" class=\"textbox examples\">\n<h3>Using the Chain Rule in a Velocity Problem<\/h3>\n<div id=\"fs-id1169736593544\" class=\"exercise\">\n<div id=\"fs-id1169736593546\" class=\"textbox\">\n<p id=\"fs-id1169736593552\">A particle moves along a coordinate axis. Its position at time [latex]t[\/latex] is given by [latex]s(t)= \\sin (2t)+ \\cos (3t).[\/latex] What is the velocity of the particle at time [latex]t=\\frac{\\pi }{6}?[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736593621\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736593621\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736593621\">To find [latex]v(t),[\/latex] the velocity of the particle at time [latex]t,[\/latex] we must differentiate [latex]s(t).[\/latex] Thus,<\/p>\n<div id=\"fs-id1169736593661\" class=\"equation unnumbered\">[latex]v(t)={s}^{\\prime }(t)=2 \\cos (2t)-3 \\sin (3t).[\/latex]<\/div>\n<p id=\"fs-id1169739266652\">Substituting [latex]t=\\frac{\\pi }{6}[\/latex] into [latex]v(t),[\/latex] we obtain [latex]v(\\frac{\\pi }{6})=-2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739266711\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169739266714\" class=\"exercise\">\n<div id=\"fs-id1169739266717\" class=\"textbox\">\n<p id=\"fs-id1169739266719\">A particle moves along a coordinate axis. Its position at time [latex]t[\/latex] is given by [latex]s(t)= \\sin (4t).[\/latex] Find its acceleration at time [latex]t.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739266765\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739266765\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739266765\">[latex]a(t)=-16 \\sin (4t)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169736655160\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169736655167\">Acceleration is the second derivative of position.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736655174\" class=\"bc-section section\">\n<h2>Proof<\/h2>\n<p id=\"fs-id1169736655179\">At this point, we present a very informal proof of the chain rule. For simplicity\u2019s sake we ignore certain issues: For example, we assume that [latex]g(x)\\ne g(a)[\/latex] for [latex]x\\ne a[\/latex] in some open interval containing [latex]a.[\/latex] We begin by applying the limit definition of the derivative to the function [latex]h(x)[\/latex] to obtain [latex]{h}^{\\prime }(a)\\text{:}[\/latex]<\/p>\n<div id=\"fs-id1169736655258\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(a)=\\underset{x\\to a}{\\text{lim}}\\frac{f(g(x))-f(g(a))}{x-a}.[\/latex]<\/div>\n<p id=\"fs-id1169739305452\">Rewriting, we obtain<\/p>\n<div id=\"fs-id1169739305455\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(a)=\\underset{x\\to a}{\\text{lim}}\\frac{f(g(x))-f(g(a))}{g(x)-g(a)}\u00b7\\frac{g(x)-g(a)}{x-a}.[\/latex]<\/div>\n<p id=\"fs-id1169739305587\">Although it is clear that<\/p>\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\text{lim}}\\frac{g(x)-g(a)}{x-a}={g}^{\\prime }(a),[\/latex]<\/div>\n<p id=\"fs-id1169736662280\">it is not obvious that<\/p>\n<div id=\"fs-id1169736662283\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\text{lim}}\\frac{f(g(x))-f(g(a))}{g(x)-g(a)}={f}^{\\prime }(g(a)).[\/latex]<\/div>\n<p id=\"fs-id1169736662391\">To see that this is true, first recall that since [latex]g[\/latex] is differentiable at [latex]a,g[\/latex] is also continuous at [latex]a.[\/latex] Thus,<\/p>\n<div id=\"fs-id1169736662414\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\text{lim}}g(x)=g(a).[\/latex]<\/div>\n<p id=\"fs-id1169739303393\">Next, make the substitution [latex]y=g(x)[\/latex] and [latex]b=g(a)[\/latex] and use change of variables in the limit to obtain<\/p>\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\text{lim}}\\frac{f(g(x))-f(g(a))}{g(x)-g(a)}=\\underset{y\\to b}{\\text{lim}}\\frac{f(y)-f(b)}{y-b}={f}^{\\prime }(b)={f}^{\\prime }(g(a)).[\/latex]<\/div>\n<p id=\"fs-id1169739335832\">Finally,<\/p>\n<div id=\"fs-id1169739335835\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(a)=\\underset{x\\to a}{\\text{lim}}\\frac{f(g(x))-f(g(a))}{g(x)-g(a)}\u00b7\\frac{g(x)-g(a)}{x-a}={f}^{\\prime }(g(a)){g}^{\\prime }(a).[\/latex]<\/div>\n<p>\u25a1<\/p>\n<div id=\"fs-id1169736613849\" class=\"textbox examples\">\n<h3>Using the Chain Rule with Functional Values<\/h3>\n<div id=\"fs-id1169736613851\" class=\"exercise\">\n<div id=\"fs-id1169736613854\" class=\"textbox\">\n<p id=\"fs-id1169736613859\">Let [latex]h(x)=f(g(x)).[\/latex] If [latex]g(1)=4,{g}^{\\prime }(1)=3,[\/latex] and [latex]{f}^{\\prime }(4)=7,[\/latex] find [latex]{h}^{\\prime }(1).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736613978\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736613978\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736613978\">Use the chain rule, then substitute.<\/p>\n<div id=\"fs-id1169736613982\" class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill {h}^{\\prime }(1)& ={f}^{\\prime }(g(1)){g}^{\\prime }(1)\\hfill & & & \\text{Apply the chain rule.}\\hfill \\\\ & ={f}^{\\prime }(4)\u00b73\\hfill & & & \\text{Substitute}g(1)=4\\text{ and }{g}^{\\prime }(1)=3.\\hfill \\\\ & =7\u00b73\\hfill & & & \\text{Substitute}f\\prime (4)=7.\\hfill \\\\ & =21\\hfill & & & \\text{Simplify.}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739341441\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169739341444\" class=\"exercise\">\n<div id=\"fs-id1169739341446\" class=\"textbox\">\n<p id=\"fs-id1169739341448\">Given [latex]h(x)=f(g(x)).[\/latex] If [latex]g(2)=-3,{g}^{\\prime }(2)=4,[\/latex] and [latex]{f}^{\\prime }(-3)=7,[\/latex] find [latex]{h}^{\\prime }(2).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739353312\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739353312\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739353312\">28<\/p>\n<\/div>\n<div id=\"fs-id1169739353320\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169739353326\">Follow <a class=\"autogenerated-content\" href=\"#fs-id1169736593542\">(Figure)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739353337\" class=\"bc-section section\">\n<h1>The Chain Rule Using Leibniz\u2019s Notation<\/h1>\n<p id=\"fs-id1169739353342\">As with other derivatives that we have seen, we can express the chain rule using Leibniz\u2019s notation. This notation for the chain rule is used heavily in physics applications.<\/p>\n<p id=\"fs-id1169739353347\">[latex]\\text{ For }h(x)=f(g(x)),[\/latex] let [latex]u=g(x)[\/latex] and [latex]y=h(x)=g(u).[\/latex] Thus,<\/p>\n<div class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)=\\frac{dy}{dx},{f}^{\\prime }(g(x))={f}^{\\prime }(u)=\\frac{dy}{du}\\text{ and }{g}^{\\prime }(x)=\\frac{du}{dx}.[\/latex]<\/div>\n<p id=\"fs-id1169739307916\">Consequently,<\/p>\n<div id=\"fs-id1169739307919\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}={h}^{\\prime }(x)={f}^{\\prime }(g(x)){g}^{\\prime }(x)=\\frac{dy}{du}\u00b7\\frac{du}{dx}.[\/latex]<\/div>\n<div id=\"fs-id1169739308016\" class=\"textbox key-takeaways\">\n<h3>Rule: Chain Rule Using Leibniz\u2019s Notation<\/h3>\n<p id=\"fs-id1169739264124\">If [latex]y[\/latex] is a function of [latex]u,[\/latex] and [latex]u[\/latex] is a function of [latex]x,[\/latex] then<\/p>\n<div id=\"fs-id1169739264151\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=\\frac{dy}{du}\u00b7\\frac{du}{dx}.[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1169739264200\" class=\"textbox examples\">\n<h3>Taking a Derivative Using Leibniz\u2019s Notation, Example 1<\/h3>\n<div id=\"fs-id1169739264202\" class=\"exercise\">\n<div id=\"fs-id1169739264204\" class=\"textbox\">\n<p id=\"fs-id1169739264209\">Find the derivative of [latex]y={(\\frac{x}{3x+2})}^{5}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739264248\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739264248\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739264248\">First, let [latex]u=\\frac{x}{3x+2}.[\/latex] Thus, [latex]y={u}^{5}.[\/latex] Next, find [latex]\\frac{du}{dx}[\/latex] and [latex]\\frac{dy}{du}.[\/latex] Using the quotient rule,<\/p>\n<div id=\"fs-id1169739264320\" class=\"equation unnumbered\">[latex]\\frac{du}{dx}=\\frac{2}{{(3x+2)}^{2}}[\/latex]<\/div>\n<p id=\"fs-id1169739289304\">and<\/p>\n<div id=\"fs-id1169739289307\" class=\"equation unnumbered\">[latex]\\frac{dy}{du}=5{u}^{4}.[\/latex]<\/div>\n<p id=\"fs-id1169739289337\">Finally, we put it all together.<\/p>\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill \\frac{dy}{dx}& =\\frac{dy}{du}\u00b7\\frac{du}{dx}\\hfill & & & \\text{Apply the chain rule.}\\hfill \\\\ & =5{u}^{4}\u00b7\\frac{2}{{(3x+2)}^{2}}\\hfill & & & \\text{Substitute}\\frac{dy}{du}=5{u}^{4}\\text{ and }\\frac{du}{dx}=\\frac{2}{{(3x+2)}^{2}}.\\hfill \\\\ & =5{(\\frac{x}{3x+2})}^{4}\u00b7\\frac{2}{{(3x+2)}^{2}}\\hfill & & & \\text{Substitute}u=\\frac{x}{3x+2}.\\hfill \\\\ & =\\frac{10{x}^{4}}{{(3x+2)}^{6}}\\hfill & & & \\text{Simplify.}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1169736592575\">It is important to remember that, when using the Leibniz form of the chain rule, the final answer must be expressed entirely in terms of the original variable given in the problem.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736592583\" class=\"textbox examples\">\n<h3>Taking a Derivative Using Leibniz\u2019s Notation, Example 2<\/h3>\n<div id=\"fs-id1169736592585\" class=\"exercise\">\n<div id=\"fs-id1169736592587\" class=\"textbox\">\n<p id=\"fs-id1169736592592\">Find the derivative of [latex]y= \\tan (4{x}^{2}-3x+1).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736661260\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736661260\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736661260\">First, let [latex]u=4{x}^{2}-3x+1.[\/latex] Then [latex]y= \\tan u.[\/latex] Next, find [latex]\\frac{du}{dx}[\/latex] and [latex]\\frac{dy}{du}\\text{:}[\/latex]<\/p>\n<div id=\"fs-id1169736661338\" class=\"equation unnumbered\">[latex]\\frac{du}{dx}=8x-3\\text{ and }\\frac{dy}{du}={ \\sec }^{2}u.[\/latex]<\/div>\n<p id=\"fs-id1169736661398\">Finally, we put it all together.<\/p>\n<div id=\"fs-id1169736661401\" class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill \\frac{dy}{dx}& =\\frac{dy}{du}\u00b7\\frac{du}{dx}\\hfill & & & \\text{Apply the chain rule.}\\hfill \\\\ & ={ \\sec }^{2}u\u00b7(8x-3)\\hfill & & & \\text{Use}\\frac{du}{dx}=8x-3\\text{ and }\\frac{dy}{du}={ \\sec }^{2}u.\\hfill \\\\ & ={ \\sec }^{2}(4{x}^{2}-3x+1)\u00b7(8x-3)\\hfill & & & \\text{Substitute}u=4{x}^{2}-3x+1.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739302081\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169739302086\" class=\"exercise\">\n<div id=\"fs-id1169739302088\" class=\"textbox\">\n<p id=\"fs-id1169739302090\">Use Leibniz\u2019s notation to find the derivative of [latex]y= \\cos ({x}^{3}).[\/latex] Make sure that the final answer is expressed entirely in terms of the variable [latex]x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736594031\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736594031\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736594031\">[latex]\\frac{dy}{dx}=-3{x}^{2} \\sin ({x}^{3})[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169736594073\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169736594079\">Let [latex]u={x}^{3}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736594101\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1169736594108\">\n<li>The chain rule allows us to differentiate compositions of two or more functions. It states that for [latex]h(x)=f(g(x)),[\/latex]\n<div id=\"fs-id1169736594153\" class=\"equation unnumbered\">[latex]{h}^{\\prime }(x)={f}^{\\prime }(g(x)){g}^{\\prime }(x).[\/latex]<\/div>\n<p>In Leibniz\u2019s notation this rule takes the form<\/p>\n<div id=\"fs-id1169737159954\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=\\frac{dy}{du}\u00b7\\frac{du}{dx}.[\/latex]<\/div>\n<\/li>\n<li>We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them.<\/li>\n<li>The chain rule combines with the power rule to form a new rule:\n<div id=\"fs-id1169737470447\" class=\"equation unnumbered\">[latex]\\text{ If }h(x)={(g(x))}^{n},\\text{then}{h}^{\\prime }(x)=n{(g(x))}^{n-1}{g}^{\\prime }(x).[\/latex]<\/div>\n<\/li>\n<li>When applied to the composition of three functions, the chain rule can be expressed as follows: If [latex]h(x)=f(g(k(x))),[\/latex] then [latex]{h}^{\\prime }(x)={f}^{\\prime }(g(k(x)){g}^{\\prime }(k(x)){k}^{\\prime }(x).[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169736654969\" class=\"key-equations\">\n<h1>Key Equations<\/h1>\n<ul id=\"fs-id1169736654976\">\n<li><strong>The chain rule<\/strong><br \/>\n[latex]{h}^{\\prime }(x)={f}^{\\prime }(g(x)){g}^{\\prime }(x)[\/latex]<\/li>\n<li><strong>The power rule for functions<\/strong><br \/>\n[latex]{h}^{\\prime }(x)=n{(g(x))}^{n-1}{g}^{\\prime }(x)[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169736655110\" class=\"textbox exercises\">\n<p id=\"fs-id1169736655114\">For the following exercises, given [latex]y=f(u)[\/latex] and [latex]u=g(x),[\/latex] find [latex]\\frac{dy}{dx}[\/latex] by using Leibniz\u2019s notation for the chain rule: [latex]\\frac{dy}{dx}=\\frac{dy}{du}\\frac{du}{dx}.[\/latex]<\/p>\n<div id=\"fs-id1169736659368\" class=\"exercise\">\n<div id=\"fs-id1169736659370\" class=\"textbox\">\n<p id=\"fs-id1169736659372\">[latex]y=3u-6,u=2{x}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736659427\" class=\"exercise\">\n<div id=\"fs-id1169736659430\" class=\"textbox\">\n<p id=\"fs-id1169736659432\">[latex]y=6{u}^{3},u=7x-4[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736659466\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736659466\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736659466\">[latex]18{u}^{2}\u00b77=18{(7x-4)}^{2}\u00b77[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739281036\" class=\"exercise\">\n<div id=\"fs-id1169739281038\" class=\"textbox\">\n<p id=\"fs-id1169739281040\">[latex]y= \\sin u,u=5x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739281115\" class=\"exercise\">\n<div id=\"fs-id1169739281117\" class=\"textbox\">\n<p id=\"fs-id1169739281119\">[latex]y= \\cos u,u=\\frac{\\text{\u2212}x}{8}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739281153\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739281153\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739281153\">[latex]\\text{\u2212} \\sin u\u00b7\\frac{-1}{8}=\\text{\u2212} \\sin (\\frac{\\text{\u2212}x}{8})\u00b7\\frac{-1}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739281207\" class=\"exercise\">\n<div id=\"fs-id1169739281210\" class=\"textbox\">\n<p id=\"fs-id1169739281212\">[latex]y= \\tan u,u=9x+2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739262398\" class=\"exercise\">\n<div id=\"fs-id1169739262400\" class=\"textbox\">\n<p id=\"fs-id1169739262402\">[latex]y=\\sqrt{4u+3},u={x}^{2}-6x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739262442\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739262442\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739262442\">[latex]\\frac{8x-24}{2\\sqrt{4u+3}}=\\frac{4x-12}{\\sqrt{4{x}^{2}-24x+3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169739262507\">For each of the following exercises,<\/p>\n<ol id=\"fs-id1169739262511\" style=\"list-style-type: lower-alpha\">\n<li>decompose each function in the form [latex]y=f(u)[\/latex] and [latex]u=g(x),[\/latex] and<\/li>\n<li>find [latex]\\frac{dy}{dx}[\/latex] as a function of [latex]x.[\/latex]<\/li>\n<\/ol>\n<div id=\"fs-id1169736602724\" class=\"exercise\">\n<div id=\"fs-id1169736602726\" class=\"textbox\">\n<p id=\"fs-id1169736602729\">[latex]y={(3x-2)}^{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736602828\" class=\"exercise\">\n<div id=\"fs-id1169736602830\" class=\"textbox\">\n<p id=\"fs-id1169736602832\">[latex]y={(3{x}^{2}+1)}^{3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736602866\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736602866\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736602866\">a. [latex]u=3{x}^{2}+1;[\/latex] b. [latex]18x{(3{x}^{2}+1)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736602922\" class=\"exercise\">\n<div id=\"fs-id1169736602924\" class=\"textbox\">\n<p id=\"fs-id1169736588867\">[latex]y={ \\sin }^{5}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736588943\" class=\"exercise\">\n<div id=\"fs-id1169736588945\" class=\"textbox\">\n<p id=\"fs-id1169736588947\">[latex]y={(\\frac{x}{7}+\\frac{7}{x})}^{7}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736588982\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736588982\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736588982\">a. [latex]f(u)={u}^{7},u=\\frac{x}{7}+\\frac{7}{x};[\/latex] b. [latex]7{(\\frac{x}{7}+\\frac{7}{x})}^{6}\u00b7(\\frac{1}{7}-\\frac{7}{{x}^{2}})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736589083\" class=\"exercise\">\n<div id=\"fs-id1169736589085\" class=\"textbox\">\n<p id=\"fs-id1169736589087\">[latex]y= \\tan ( \\sec x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739251116\" class=\"exercise\">\n<div id=\"fs-id1169739251118\" class=\"textbox\">\n<p id=\"fs-id1169739251120\">[latex]y= \\csc (\\pi x+1)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739251149\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739251149\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739251149\">a. [latex]f(u)= \\csc u,u=\\pi x+1;[\/latex] b. [latex]\\text{\u2212}\\pi \\csc (\\pi x+1)\u00b7 \\cot (\\pi x+1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739280322\" class=\"exercise\">\n<div id=\"fs-id1169739280325\" class=\"textbox\">\n<p id=\"fs-id1169739280327\">[latex]y={ \\cot }^{2}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739280417\" class=\"exercise\">\n<div id=\"fs-id1169739280419\" class=\"textbox\">\n<p id=\"fs-id1169739280421\">[latex]y=-6{ \\sin }^{-3}x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739280445\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739280445\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739280445\">a. [latex]f(u)=-6{u}^{-3},u= \\sin x,[\/latex] b. [latex]18{ \\sin }^{-4}x\u00b7 \\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169739280513\">For the following exercises, find [latex]\\frac{dy}{dx}[\/latex] for each function.<\/p>\n<div id=\"fs-id1169736653150\" class=\"exercise\">\n<div id=\"fs-id1169736653152\" class=\"textbox\">\n<p id=\"fs-id1169736653155\">[latex]y={(3{x}^{2}+3x-1)}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736653250\" class=\"exercise\">\n<div id=\"fs-id1169736653252\" class=\"textbox\">\n<p id=\"fs-id1169736653254\">[latex]y={(5-2x)}^{-2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736653286\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736653286\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736653286\">[latex]\\frac{4}{{(5-2x)}^{3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736653316\" class=\"exercise\">\n<div id=\"fs-id1169736653318\" class=\"textbox\">\n<p id=\"fs-id1169736653320\">[latex]y={ \\cos }^{3}(\\pi x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739195140\" class=\"exercise\">\n<div id=\"fs-id1169739195142\" class=\"textbox\">\n<p id=\"fs-id1169739195144\">[latex]y={(2{x}^{3}-{x}^{2}+6x+1)}^{3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739195192\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739195192\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739195192\">[latex]6{(2{x}^{3}-{x}^{2}+6x+1)}^{2}(3{x}^{2}-x+3)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739195259\" class=\"exercise\">\n<div id=\"fs-id1169739195261\" class=\"textbox\">\n<p id=\"fs-id1169739195263\">[latex]y=\\frac{1}{{ \\sin }^{2}(x)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739195337\" class=\"exercise\">\n<div id=\"fs-id1169739195339\" class=\"textbox\">\n<p id=\"fs-id1169739195341\">[latex]y={( \\tan x+ \\sin x)}^{-3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738989496\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738989496\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738989496\">[latex]-3{( \\tan x+ \\sin x)}^{-4}\u00b7({ \\sec }^{2}x+ \\cos x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738989560\" class=\"exercise\">\n<div id=\"fs-id1169738989562\" class=\"textbox\">\n<p id=\"fs-id1169738989564\">[latex]y={x}^{2}{ \\cos }^{4}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738989642\" class=\"exercise\">\n<div id=\"fs-id1169738989645\" class=\"textbox\">\n<p id=\"fs-id1169738989647\">[latex]y= \\sin ( \\cos 7x)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738989676\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738989676\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738989676\">[latex]-7 \\cos ( \\cos 7x)\u00b7 \\sin 7x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736585006\" class=\"exercise\">\n<div id=\"fs-id1169736585009\" class=\"textbox\">\n<p id=\"fs-id1169736585011\">[latex]y=\\sqrt{6+ \\sec \\pi {x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736585115\" class=\"exercise\">\n<div id=\"fs-id1169736585117\" class=\"textbox\">\n<p id=\"fs-id1169736585119\">[latex]y={ \\cot }^{3}(4x+1)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736585152\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736585152\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736585152\">[latex]-12{ \\cot }^{2}(4x+1)\u00b7{ \\csc }^{2}(4x+1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738988193\" class=\"exercise\">\n<div id=\"fs-id1169738988195\" class=\"textbox\">\n<p id=\"fs-id1169738988197\">Let [latex]y={\\left[f(x)\\right]}^{3}[\/latex] and suppose that [latex]{f}^{\\prime }(1)=4[\/latex] and [latex]\\frac{dy}{dx}=10[\/latex] for [latex]x=1.[\/latex] Find [latex]f(1).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738988313\" class=\"exercise\">\n<div id=\"fs-id1169738988315\" class=\"textbox\">\n<p id=\"fs-id1169738988317\">Let [latex]y={(f(x)+5{x}^{2})}^{4}[\/latex] and suppose that [latex]f(-1)=-4[\/latex] and [latex]\\frac{dy}{dx}=3[\/latex] when [latex]x=-1.[\/latex] Find [latex]{f}^{\\prime }(-1)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739374485\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739374485\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739374485\">[latex]10\\frac{3}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739374499\" class=\"exercise\">\n<div id=\"fs-id1169739374501\" class=\"textbox\">\n<p id=\"fs-id1169739374503\">Let [latex]y={(f(u)+3x)}^{2}[\/latex] and [latex]u={x}^{3}-2x.[\/latex] If [latex]f(4)=6[\/latex] and [latex]\\frac{dy}{dx}=18[\/latex] when [latex]x=2,[\/latex] find [latex]{f}^{\\prime }(4).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739374644\" class=\"exercise\">\n<div id=\"fs-id1169739374646\" class=\"textbox\">\n<p id=\"fs-id1169739374648\"><strong>[T]<\/strong> Find the equation of the tangent line to [latex]y=\\text{\u2212} \\sin (\\frac{x}{2})[\/latex] at the origin. Use a calculator to graph the function and the tangent line together.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739374682\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739374682\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739374682\">[latex]y=\\frac{-1}{2}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738991878\" class=\"exercise\">\n<div id=\"fs-id1169738991880\" class=\"textbox\">\n<p id=\"fs-id1169738991882\"><strong>[T]<\/strong> Find the equation of the tangent line to [latex]y={(3x+\\frac{1}{x})}^{2}[\/latex] at the point [latex](1,16).[\/latex] Use a calculator to graph the function and the tangent line together.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738991955\" class=\"exercise\">\n<div id=\"fs-id1169738991958\" class=\"textbox\">\n<p id=\"fs-id1169738991960\">Find the [latex]x[\/latex]-coordinates at which the tangent line to [latex]y={(x-\\frac{6}{x})}^{8}[\/latex] is horizontal.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738991998\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738991998\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738991998\">[latex]x=\u00b1\\sqrt{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738992014\" class=\"exercise\">\n<div id=\"fs-id1169738992016\" class=\"textbox\">\n<p id=\"fs-id1169738992018\"><strong>[T]<\/strong> Find an equation of the line that is normal to [latex]g(\\theta )={ \\sin }^{2}(\\pi \\theta )[\/latex] at the point [latex](\\frac{1}{4},\\frac{1}{2}).[\/latex] Use a calculator to graph the function and the normal line together.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169736582610\">For the following exercises, use the information in the following table to find [latex]{h}^{\\prime }(a)[\/latex] at the given value for [latex]a.[\/latex]<\/p>\n<table id=\"fs-id1169736582646\" class=\"unnumbered\" summary=\"This table has five rows and five columns. The first row is a header row and it labels each column. The column headers from left to right are x, f(x), f\u2019(x), g(x), and g\u2019(x). Under the first column are the values 0, 1, 2, and 3. Under the second column are the values 2, 1, 4, and 3. Under the third column are the values 5, \u22122, 4, and \u22123. Under the fourth column are the values 0, 3, 1, and 2. Under the fifth column are g\u2019(x) are the values 2, 0, \u22121, and 3.\">\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)[\/latex]<\/th>\n<th>[latex]f\\prime (x)[\/latex]<\/th>\n<th>[latex]g(x)[\/latex]<\/th>\n<th>[latex]g\\prime (x)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>0<\/td>\n<td>2<\/td>\n<td>5<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1<\/td>\n<td>1<\/td>\n<td>\u22122<\/td>\n<td>3<\/td>\n<td>0<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2<\/td>\n<td>4<\/td>\n<td>4<\/td>\n<td>1<\/td>\n<td>\u22121<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>3<\/td>\n<td>3<\/td>\n<td>\u22123<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1169739243334\" class=\"exercise\">\n<div id=\"fs-id1169739243336\" class=\"textbox\">\n<p id=\"fs-id1169739243338\">[latex]h(x)=f(g(x));a=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739243384\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739243384\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739243384\">10<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739243389\" class=\"exercise\">\n<div id=\"fs-id1169739243391\" class=\"textbox\">\n<p id=\"fs-id1169739243393\">[latex]h(x)=g(f(x));a=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736662054\" class=\"exercise\">\n<div id=\"fs-id1169736662056\" class=\"textbox\">\n<p id=\"fs-id1169736662058\">[latex]h(x)={({x}^{4}+g(x))}^{-2};a=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736662118\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736662118\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736662118\">[latex]-\\frac{1}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736662132\" class=\"exercise\">\n<div id=\"fs-id1169736662134\" class=\"textbox\">\n<p id=\"fs-id1169736662136\">[latex]h(x)={(\\frac{f(x)}{g(x)})}^{2};a=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736662220\" class=\"exercise\">\n<div id=\"fs-id1169736662223\" class=\"textbox\">\n<p id=\"fs-id1169736662225\">[latex]h(x)=f(x+f(x));a=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736608228\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736608228\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736608228\">-4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736608237\" class=\"exercise\">\n<div id=\"fs-id1169736608239\" class=\"textbox\">\n<p id=\"fs-id1169736608241\">[latex]h(x)={(1+g(x))}^{3};a=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736608305\" class=\"exercise\">\n<div id=\"fs-id1169736608307\" class=\"textbox\">\n<p id=\"fs-id1169736608309\">[latex]h(x)=g(2+f({x}^{2}));a=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736608364\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736608364\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736608364\">-12<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736608372\" class=\"exercise\">\n<div id=\"fs-id1169736608374\" class=\"textbox\">\n<p id=\"fs-id1169736608376\">[latex]h(x)=f(g( \\sin x));a=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736608434\" class=\"exercise\">\n<div id=\"fs-id1169736592171\" class=\"textbox\">\n<p id=\"fs-id1169736592173\"><strong>[T]<\/strong> The position function of a freight train is given by [latex]s(t)=100{(t+1)}^{-2},[\/latex] with [latex]s[\/latex] in meters and [latex]t[\/latex] in seconds. At time [latex]t=6[\/latex] s, find the train\u2019s<\/p>\n<ol id=\"fs-id1169736592236\" style=\"list-style-type: lower-alpha\">\n<li>velocity and<\/li>\n<li>acceleration.<\/li>\n<li>Using a. and b. is the train speeding up or slowing down?<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736592256\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736592256\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736592256\">a. [latex]-\\frac{200}{343}[\/latex] m\/s, b. [latex]\\frac{600}{2401}[\/latex] m\/s<sup>2<\/sup>, c. The train is slowing down since velocity and acceleration have opposite signs.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736592288\" class=\"exercise\">\n<div id=\"fs-id1169736592290\" class=\"textbox\">\n<p id=\"fs-id1169736592292\"><strong>[T]<\/strong> A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where [latex]t[\/latex] is measured in seconds and [latex]s[\/latex] is in inches:<\/p>\n<p id=\"fs-id1169736592309\">[latex]s(t)=-3 \\cos (\\pi t+\\frac{\\pi }{4}).[\/latex]<\/p>\n<ol id=\"fs-id1169736592353\" style=\"list-style-type: lower-alpha\">\n<li>Determine the position of the spring at [latex]t=1.5[\/latex] s.<\/li>\n<li>Find the velocity of the spring at [latex]t=1.5[\/latex] s.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736592402\" class=\"exercise\">\n<div id=\"fs-id1169736616212\" class=\"textbox\">\n<p id=\"fs-id1169736616214\"><strong>[T]<\/strong> The total cost to produce [latex]x[\/latex] boxes of Thin Mint Girl Scout cookies is [latex]C[\/latex] dollars, where [latex]C=0.0001{x}^{3}-0.02{x}^{2}+3x+300.[\/latex] In [latex]t[\/latex] weeks production is estimated to be [latex]x=1600+100t[\/latex] boxes.<\/p>\n<ol id=\"fs-id1169736616286\" style=\"list-style-type: lower-alpha\">\n<li>Find the marginal cost [latex]{C}^{\\prime }(x).[\/latex]<\/li>\n<li>Use Leibniz\u2019s notation for the chain rule, [latex]\\frac{dC}{dt}=\\frac{dC}{dx}\u00b7\\frac{dx}{dt},[\/latex] to find the rate with respect to time [latex]t[\/latex] that the cost is changing.<\/li>\n<li>Use b. to determine how fast costs are increasing when [latex]t=2[\/latex] weeks. Include units with the answer.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736616383\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736616383\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736616383\">a. [latex]{C}^{\\prime }(x)=0.0003{x}^{2}-0.04x+3[\/latex] b. [latex]\\frac{dC}{dt}=100\u00b7(0.0003{x}^{2}-0.04x+3)[\/latex] c. Approximately $90,300 per week<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736618622\" class=\"exercise\">\n<div id=\"fs-id1169736618624\" class=\"textbox\">\n<p id=\"fs-id1169736618626\"><strong>[T]<\/strong> The formula for the area of a circle is [latex]A=\\pi {r}^{2},[\/latex] where [latex]r[\/latex] is the radius of the circle. Suppose a circle is expanding, meaning that both the area [latex]A[\/latex] and the radius [latex]r[\/latex] (in inches) are expanding.<\/p>\n<ol id=\"fs-id1169736618664\" style=\"list-style-type: lower-alpha\">\n<li>Suppose [latex]r=2-\\frac{100}{{(t+7)}^{2}}[\/latex] where [latex]t[\/latex] is time in seconds. Use the chain rule [latex]\\frac{dA}{dt}=\\frac{dA}{dr}\u00b7\\frac{dr}{dt}[\/latex] to find the rate at which the area is expanding.<\/li>\n<li>Use a. to find the rate at which the area is expanding at [latex]t=4[\/latex] s.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736618829\" class=\"exercise\">\n<div id=\"fs-id1169736618831\" class=\"textbox\">\n<p id=\"fs-id1169736618834\"><strong>[T]<\/strong> The formula for the volume of a sphere is [latex]S=\\frac{4}{3}\\pi {r}^{3},[\/latex] where [latex]r[\/latex] (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.<\/p>\n<ol id=\"fs-id1169739066678\" style=\"list-style-type: lower-alpha\">\n<li>Suppose [latex]r=\\frac{1}{{(t+1)}^{2}}-\\frac{1}{12}[\/latex] where [latex]t[\/latex] is time in minutes. Use the chain rule [latex]\\frac{dS}{dt}=\\frac{dS}{dr}\u00b7\\frac{dr}{dt}[\/latex] to find the rate at which the snowball is melting.<\/li>\n<li>Use a. to find the rate at which the volume is changing at [latex]t=1[\/latex] min.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739066789\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739066789\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739066789\">a. [latex]\\frac{dS}{dt}=-\\frac{8\\pi {r}^{2}}{{(t+1)}^{3}}[\/latex] b. The volume is decreasing at a rate of [latex]-\\frac{\\pi }{36}[\/latex] ft<sup>3<\/sup>\/min.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739066858\" class=\"exercise\">\n<div id=\"fs-id1169739066860\" class=\"textbox\">\n<p id=\"fs-id1169739066862\"><strong>[T]<\/strong> The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function [latex]T(x)=94-10 \\cos \\left[\\frac{\\pi }{12}(x-2)\\right],[\/latex] where [latex]x[\/latex] is hours after midnight. Find the rate at which the temperature is changing at 4 p.m.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739296655\" class=\"exercise\">\n<div id=\"fs-id1169739296657\" class=\"textbox\">\n<p id=\"fs-id1169739296659\"><strong>[T]<\/strong> The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function [latex]D(t)=5 \\sin (\\frac{\\pi }{6}t-\\frac{7\\pi }{6})+8,[\/latex] where [latex]t[\/latex] is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739296729\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739296729\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739296729\">[latex]~2.3[\/latex] ft\/hr<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169739296745\" class=\"definition\">\n<dt>chain rule<\/dt>\n<dd id=\"fs-id1169739296750\">the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":311,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1848","chapter","type-chapter","status-publish","hentry"],"part":1777,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1848","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1848\/revisions"}],"predecessor-version":[{"id":2433,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1848\/revisions\/2433"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/parts\/1777"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1848\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/media?parent=1848"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapter-type?post=1848"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/contributor?post=1848"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/license?post=1848"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}