{"id":1990,"date":"2021-08-19T16:07:01","date_gmt":"2021-08-19T16:07:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-basics-of-differential-equations\/"},"modified":"2021-11-19T03:10:07","modified_gmt":"2021-11-19T03:10:07","slug":"skills-review-for-basics-of-differential-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-basics-of-differential-equations\/","title":{"raw":"Skills Review for Basics of Differential Equations","rendered":"Skills Review for Basics of Differential Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\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>Write function equations using given conditions<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Basics of Differential Equations section, we will verify the solutions of differential equations by taking derivatives. We will also find exact solutions of differential equations by using given conditions. These skills are reviewed here.\r\n<h2>Use the Chain Rule<\/h2>\r\n<div id=\"fs-id1169739194586\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">The Chain Rule<\/h3>\r\n\r\n<hr \/>\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\" style=\"text-align: center;\">[latex]h(x)=(f\\circ g)(x)=f(g(x))[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738949353\">is given by<\/p>\r\n\r\n<div id=\"fs-id1169738948938\" class=\"equation\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=f^{\\prime}(g(x))g^{\\prime}(x)[\/latex]<\/div>\r\n&nbsp;\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><\/div>\r\n<div id=\"fs-id1169739187558\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}=\\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex]<\/div>\r\n<\/div>\r\n<div>\r\n<p id=\"fs-id1169739096228\">Note that we often need to use the chain rule 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)=nx^{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\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=n(g(x))^{n-1}g^{\\prime}(x)[\/latex]<\/div>\r\n<div><\/div>\r\n<div id=\"fs-id1169739187734\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Power Rule for Composition of Functions<\/h3>\r\n\r\n<hr \/>\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\" style=\"text-align: center;\">[latex]h(x)=(g(x))^n[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739242349\">Then<\/p>\r\n\r\n<div id=\"fs-id1169739222795\" class=\"equation\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=n(g(x))^{n-1}g^{\\prime}(x)[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739274312\" class=\"textbook exercises\">\r\n<h3>Example: Using the Chain and Power Rules<\/h3>\r\n<p id=\"fs-id1169736589119\">Find the derivative of [latex]h(x)=\\dfrac{1}{(3x^2+1)^2}[\/latex]<\/p>\r\n[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}{(3x^2+1)^2}=(3x^2+1)^{-2}[\/latex].<\/p>\r\n<p id=\"fs-id1169739333152\">Applying the power rule with [latex]g(x)=3x^2+1[\/latex], we have<\/p>\r\n\r\n<div id=\"fs-id1169736609881\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=-2(3x^2+1)^{-3}(6x)[\/latex].<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=\\frac{-12x}{(3x^2+1)^3}[\/latex].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736662938\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169736595961\">Find the derivative of [latex]h(x)=(2x^3+2x-1)^4[\/latex]<\/p>\r\n[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(2x^3+2x-1)^3(6x^{2}+2)=8(3x^{2}+1)(2x^3+2x-1)^3[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169739274677\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169739179049\">Use the previous example with [latex]g(x)=2x^3+2x-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739302258\" class=\"textbook exercises\">\r\n<h3>Example: Using the Chain and Power Rules with a Trigonometric Function<\/h3>\r\n<p id=\"fs-id1169739273001\">Find the derivative of [latex]h(x)=\\sin^3 x[\/latex]<\/p>\r\n[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].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\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-id1169739301537\" class=\"textbook exercises\">\r\n<h3>Example: Using the Chain Rule on a Cosine Function<\/h3>\r\n<p id=\"fs-id1169739301547\">Find the derivative of [latex]h(x)= \\cos (5x^2)[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1169739111373\" class=\"equation unnumbered\">[reveal-answer q=\"847788\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"847788\"]Let [latex]g(x)=5x^2[\/latex]. Then [latex]g^{\\prime}(x)=10x[\/latex].\r\nUsing the result from the previous example, [latex]\\begin{array}{ll}h^{\\prime}(x) &amp; =-\\sin (5x^2) \\cdot 10x \\\\ &amp; =-10x \\sin (5x^2) \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739333921\" class=\"textbook exercises\">\r\n<h3>Example: Using the Chain Rule on Another Trigonometric Function<\/h3>\r\n<p id=\"fs-id1169739333931\">Find the derivative of [latex]h(x)= \\sec (4x^5+2x)[\/latex].<\/p>\r\n[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&nbsp;\r\n<p id=\"fs-id1169736615162\">In this problem, [latex]g(x)=4x^5+2x[\/latex], so we have [latex]g^{\\prime}(x)=20x^4+2[\/latex]. Therefore, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739299798\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}h^{\\prime}(x) &amp; = \\sec (4x^5+2x) \\tan (4x^5+2x)(20x^4+2) \\\\ &amp; =(20x^4+2) \\sec (4x^5+2x) \\tan (4x^5+2x) \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739188144\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739188151\">Find the derivative of [latex]h(x)= \\sin (7x+2)[\/latex].<\/p>\r\n[reveal-answer q=\"166577\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"166577\"]\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[\/hidden-answer]\r\n\r\n[reveal-answer q=\"232193\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"232193\"]\r\n\r\n[latex]h^{\\prime}(x)=7 \\cos (7x+2)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1169736610159\">We now 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 the last three examples. 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 shaded\">\r\n<h3 style=\"text-align: center;\">Using the Chain Rule with Trigonometric Functions<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1169736655841\">For all values of [latex]x[\/latex] for which the derivative is defined,<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{llll}\\frac{d}{dx}(\\sin (g(x)))= \\cos (g(x))g^{\\prime}(x) &amp; &amp; &amp; \\frac{d}{dx} \\sin u= \\cos u\\frac{du}{dx} \\\\ \\frac{d}{dx}(\\cos (g(x)))=\u2212\\sin (g(x))g^{\\prime}(x) &amp; &amp; &amp; \\frac{d}{dx} \\cos u=\u2212\\sin u\\frac{du}{dx} \\\\ \\frac{d}{dx}(\\tan (g(x)))= \\sec^2 (g(x))g^{\\prime}(x) &amp; &amp; &amp; \\frac{d}{dx} \\tan u=\\sec^2 u\\frac{du}{dx} \\\\ \\frac{d}{dx}(\\cot (g(x)))=\u2212\\csc^2 (g(x))g^{\\prime}(x) &amp; &amp; &amp; \\frac{d}{dx} \\cot u=\u2212\\csc^2 u\\frac{du}{dx} \\\\ \\frac{d}{dx}(\\sec (g(x)))= \\sec (g(x)) \\tan (g(x))g^{\\prime}(x) &amp; &amp; &amp; \\frac{d}{dx} \\sec u= \\sec u \\tan u\\frac{du}{dx} \\\\ \\frac{d}{dx}(\\csc (g(x)))=\u2212\\csc (g(x)) \\cot (g(x))g^{\\prime}(x) &amp; &amp; &amp; \\frac{d}{dx} \\csc u=\u2212\\csc u \\cot u\\frac{du}{dx} \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>Write Function Equations Using Given Conditions<\/h2>\r\nSometimes, to find a missing value in a function equation, you will be given an input of the function and the corresponding output. You will then plug this input and output into the function equation and find the missing value.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing a Function Equation from given conditions<\/h3>\r\nGiven [latex]f(2)=-1[\/latex], find the unknown value c in the function equation [latex]f(x)=3x^3-4x^2-x+c[\/latex].\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"338564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"338564\"]\r\n\r\nTo find c, we use the fact that [latex]f(2)=-1[\/latex], that is, the function's value is -1 when [latex]x=2[\/latex].\r\n\r\n[latex]\\begin{array}{l}-1=3(2)^3-4(2)^2-2+c\\hfill \\\\ -1=3(8)-4(4)-2+c\\hfill \\\\ -1=24-16-2+c\\hfill \\\\ -1=6+c\\hfill \\\\ -7=c \\end{array}[\/latex]\r\n\r\nThe function equation is\u00a0[latex]f(x)=3x^3-4x^2-x-7[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven [latex]f(1)=5[\/latex], find the unknown value c in the function equation [latex]f(x)=-2x^2+3x+c[\/latex].\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"338565\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"338565\"]\r\n\r\nThe function equation is [latex]f(x)=-2x^2+3x+4[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\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>Write function equations using given conditions<\/li>\n<\/ul>\n<\/div>\n<p>In the Basics of Differential Equations section, we will verify the solutions of differential equations by taking derivatives. We will also find exact solutions of differential equations by using given conditions. These skills are reviewed here.<\/p>\n<h2>Use the Chain Rule<\/h2>\n<div id=\"fs-id1169739194586\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">The Chain Rule<\/h3>\n<hr \/>\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\" style=\"text-align: center;\">[latex]h(x)=(f\\circ g)(x)=f(g(x))[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738949353\">is given by<\/p>\n<div id=\"fs-id1169738948938\" class=\"equation\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=f^{\\prime}(g(x))g^{\\prime}(x)[\/latex]<\/div>\n<p>&nbsp;<\/p>\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><\/div>\n<div id=\"fs-id1169739187558\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}=\\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex]<\/div>\n<\/div>\n<div>\n<p id=\"fs-id1169739096228\">Note that we often need to use the chain rule 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)=nx^{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\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=n(g(x))^{n-1}g^{\\prime}(x)[\/latex]<\/div>\n<div><\/div>\n<div id=\"fs-id1169739187734\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Power Rule for Composition of Functions<\/h3>\n<hr \/>\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\" style=\"text-align: center;\">[latex]h(x)=(g(x))^n[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739242349\">Then<\/p>\n<div id=\"fs-id1169739222795\" class=\"equation\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=n(g(x))^{n-1}g^{\\prime}(x)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1169739274312\" class=\"textbook exercises\">\n<h3>Example: Using the Chain and Power Rules<\/h3>\n<p id=\"fs-id1169736589119\">Find the derivative of [latex]h(x)=\\dfrac{1}{(3x^2+1)^2}[\/latex]<\/p>\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}{(3x^2+1)^2}=(3x^2+1)^{-2}[\/latex].<\/p>\n<p id=\"fs-id1169739333152\">Applying the power rule with [latex]g(x)=3x^2+1[\/latex], we have<\/p>\n<div id=\"fs-id1169736609881\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=-2(3x^2+1)^{-3}(6x)[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169736655793\">Rewriting back to the original form gives us<\/p>\n<div id=\"fs-id1169739008104\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=\\frac{-12x}{(3x^2+1)^3}[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736662938\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169736595961\">Find the derivative of [latex]h(x)=(2x^3+2x-1)^4[\/latex]<\/p>\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(2x^3+2x-1)^3(6x^{2}+2)=8(3x^{2}+1)(2x^3+2x-1)^3[\/latex]<\/p>\n<div id=\"fs-id1169739274677\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169739179049\">Use the previous example with [latex]g(x)=2x^3+2x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739302258\" class=\"textbook exercises\">\n<h3>Example: Using the Chain and Power Rules with a Trigonometric Function<\/h3>\n<p id=\"fs-id1169739273001\">Find the derivative of [latex]h(x)=\\sin^3 x[\/latex]<\/p>\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<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-id1169739301537\" class=\"textbook exercises\">\n<h3>Example: Using the Chain Rule on a Cosine Function<\/h3>\n<p id=\"fs-id1169739301547\">Find the derivative of [latex]h(x)= \\cos (5x^2)[\/latex].<\/p>\n<div id=\"fs-id1169739111373\" class=\"equation unnumbered\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q847788\">Show Solution<\/span><\/p>\n<div id=\"q847788\" class=\"hidden-answer\" style=\"display: none\">Let [latex]g(x)=5x^2[\/latex]. Then [latex]g^{\\prime}(x)=10x[\/latex].<br \/>\nUsing the result from the previous example, [latex]\\begin{array}{ll}h^{\\prime}(x) & =-\\sin (5x^2) \\cdot 10x \\\\ & =-10x \\sin (5x^2) \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739333921\" class=\"textbook exercises\">\n<h3>Example: Using the Chain Rule on Another Trigonometric Function<\/h3>\n<p id=\"fs-id1169739333931\">Find the derivative of [latex]h(x)= \\sec (4x^5+2x)[\/latex].<\/p>\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>&nbsp;<\/p>\n<p id=\"fs-id1169736615162\">In this problem, [latex]g(x)=4x^5+2x[\/latex], so we have [latex]g^{\\prime}(x)=20x^4+2[\/latex]. Therefore, we obtain<\/p>\n<div id=\"fs-id1169739299798\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}h^{\\prime}(x) & = \\sec (4x^5+2x) \\tan (4x^5+2x)(20x^4+2) \\\\ & =(20x^4+2) \\sec (4x^5+2x) \\tan (4x^5+2x) \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739188144\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739188151\">Find the derivative of [latex]h(x)= \\sin (7x+2)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q166577\">Hint<\/span><\/p>\n<div id=\"q166577\" class=\"hidden-answer\" style=\"display: none\">\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q232193\">Show Solution<\/span><\/p>\n<div id=\"q232193\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]h^{\\prime}(x)=7 \\cos (7x+2)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169736610159\">We now 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 the last three examples. 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 shaded\">\n<h3 style=\"text-align: center;\">Using the Chain Rule with Trigonometric Functions<\/h3>\n<hr \/>\n<p id=\"fs-id1169736655841\">For all values of [latex]x[\/latex] for which the derivative is defined,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{llll}\\frac{d}{dx}(\\sin (g(x)))= \\cos (g(x))g^{\\prime}(x) & & & \\frac{d}{dx} \\sin u= \\cos u\\frac{du}{dx} \\\\ \\frac{d}{dx}(\\cos (g(x)))=\u2212\\sin (g(x))g^{\\prime}(x) & & & \\frac{d}{dx} \\cos u=\u2212\\sin u\\frac{du}{dx} \\\\ \\frac{d}{dx}(\\tan (g(x)))= \\sec^2 (g(x))g^{\\prime}(x) & & & \\frac{d}{dx} \\tan u=\\sec^2 u\\frac{du}{dx} \\\\ \\frac{d}{dx}(\\cot (g(x)))=\u2212\\csc^2 (g(x))g^{\\prime}(x) & & & \\frac{d}{dx} \\cot u=\u2212\\csc^2 u\\frac{du}{dx} \\\\ \\frac{d}{dx}(\\sec (g(x)))= \\sec (g(x)) \\tan (g(x))g^{\\prime}(x) & & & \\frac{d}{dx} \\sec u= \\sec u \\tan u\\frac{du}{dx} \\\\ \\frac{d}{dx}(\\csc (g(x)))=\u2212\\csc (g(x)) \\cot (g(x))g^{\\prime}(x) & & & \\frac{d}{dx} \\csc u=\u2212\\csc u \\cot u\\frac{du}{dx} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<h2>Write Function Equations Using Given Conditions<\/h2>\n<p>Sometimes, to find a missing value in a function equation, you will be given an input of the function and the corresponding output. You will then plug this input and output into the function equation and find the missing value.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Function Equation from given conditions<\/h3>\n<p>Given [latex]f(2)=-1[\/latex], find the unknown value c in the function equation [latex]f(x)=3x^3-4x^2-x+c[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q338564\">Show Solution<\/span><\/p>\n<div id=\"q338564\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find c, we use the fact that [latex]f(2)=-1[\/latex], that is, the function&#8217;s value is -1 when [latex]x=2[\/latex].<\/p>\n<p>[latex]\\begin{array}{l}-1=3(2)^3-4(2)^2-2+c\\hfill \\\\ -1=3(8)-4(4)-2+c\\hfill \\\\ -1=24-16-2+c\\hfill \\\\ -1=6+c\\hfill \\\\ -7=c \\end{array}[\/latex]<\/p>\n<p>The function equation is\u00a0[latex]f(x)=3x^3-4x^2-x-7[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given [latex]f(1)=5[\/latex], find the unknown value c in the function equation [latex]f(x)=-2x^2+3x+c[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q338565\">Show Solution<\/span><\/p>\n<div id=\"q338565\" class=\"hidden-answer\" style=\"display: none\">\n<p>The function equation is [latex]f(x)=-2x^2+3x+4[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1990\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/precalculus\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1990","chapter","type-chapter","status-publish","hentry"],"part":536,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1990","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1990\/revisions"}],"predecessor-version":[{"id":2553,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1990\/revisions\/2553"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/536"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1990\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1990"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1990"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1990"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1990"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}