{"id":4090,"date":"2022-04-14T18:15:53","date_gmt":"2022-04-14T18:15:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-calculus-of-vector-valued-functions\/"},"modified":"2022-06-30T17:09:15","modified_gmt":"2022-06-30T17:09:15","slug":"skills-review-for-calculus-of-vector-valued-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-calculus-of-vector-valued-functions\/","title":{"raw":"Skills Review for Calculus of Vector-Valued Functions","rendered":"Skills Review for Calculus of Vector-Valued Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Apply basic derivative rules<\/li>\r\n \t<li>Use the product rule for finding the derivative of a product of functions<\/li>\r\n \t<li>Use the quotient rule for finding the derivative of a quotient of functions<\/li>\r\n \t<li>Apply the chain rule together with the power and product rule<\/li>\r\n \t<li>Evaluate indefinite integrals<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Calculus of Vector-Valued Functions section, we will learn how to differentiate and integrate vector-valued functions. Here we will review various derivative rules and integration techniques.\r\n<h2>Basic Derivative Rules<\/h2>\r\n<p id=\"fs-id1169738835554\">We first apply the limit definition of the derivative to find the derivative of the constant function, [latex]f(x)=c[\/latex]. For this function, both [latex]f(x)=c[\/latex] and [latex]f(x+h)=c[\/latex], so we obtain the following result:<\/p>\r\n\r\n<div id=\"fs-id1169738850732\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}f^{\\prime}(x) &amp; =\\underset{h\\to 0}{\\lim}\\dfrac{f(x+h)-f(x)}{h} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\dfrac{c-c}{h} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\dfrac{0}{h} \\\\ &amp; =\\underset{h\\to 0}{\\lim}0=0 \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738850205\">The rule for differentiating constant functions is called the <strong>constant rule<\/strong>. It states that the derivative of a constant function is zero; that is, since a constant function is a horizontal line, the slope, or the rate of change, of a constant function is 0. We restate this rule in the following theorem.<\/p>\r\n\r\n\r\n<div id=\"fs-id1169738828810\" class=\"bc-section section\">\r\n<div id=\"fs-id1169739030384\" class=\"bc-section section\">\r\n<div id=\"fs-id1169738955425\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">The Constant Rule<\/h3>\r\n\r\n<hr>\r\n<p id=\"fs-id1169738878658\">Let [latex]c[\/latex] be a constant.<\/p>\r\n<p id=\"fs-id1169738853363\" style=\"text-align: center;\">If [latex]f(x)=c[\/latex], then [latex]f^{\\prime}(c)=0[\/latex]<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1169739024163\">Alternatively, we may express this rule as<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(c)=0[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739274547\" class=\"textbook exercises\">\r\n<h3>Example: Applying the Constant Rule<\/h3>\r\n<p id=\"fs-id1169738824417\">Find the derivative of [latex]f(x)=8[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169738865666\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738865666\"]\r\n<p id=\"fs-id1169738865666\">This is just a one-step application of the rule:<\/p>\r\n\r\n<div id=\"fs-id1169738875468\" class=\"equation unnumbered\">[latex]f^{\\prime}(8)=0[\/latex].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738954922\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169736614166\">Find the derivative of [latex]g(x)=-3[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169738853102\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738853102\"]\r\n<p id=\"fs-id1169738853102\">0<\/p>\r\n\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169738907507\">Use the preceding example as a guide.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1169739006200\">We have shown that<\/p>\r\n\r\n<div id=\"fs-id1169739234394\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\dfrac{d}{dx}\\left(x^2\\right)=2x[\/latex]\u00a0 \u00a0and\u00a0 \u00a0[latex]\\dfrac{d}{dx}\\left(x^{\\frac{1}{2}}\\right)=\\dfrac{1}{2}x^{\u2212\\frac{1}{2}}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738969429\">At this point, you might see a pattern beginning to develop for derivatives of the form [latex]\\frac{d}{dx}(x^n)[\/latex]. We continue our examination of derivative formulas by differentiating power functions of the form [latex]f(x)=x^n[\/latex] where [latex]n[\/latex] is a positive integer. We develop formulas for derivatives of this type of function in stages, beginning with positive integer powers. Before stating and proving the general rule for derivatives of functions of this form, we take a look at a specific case, [latex]\\frac{d}{dx}(x^3)[\/latex].<\/p>\r\n\r\n<p id=\"fs-id1169736619689\">As we shall see, the procedure for finding the derivative of the general form [latex]f(x)=x^n[\/latex] is very similar. Although it is often unwise to draw general conclusions from specific examples, we note that when we differentiate [latex]f(x)=x^3[\/latex], the power on [latex]x[\/latex] becomes the coefficient of [latex]x^2[\/latex] in the derivative and the power on [latex]x[\/latex] in the derivative decreases by 1. The following theorem states that this\u00a0<strong>power rule<\/strong> holds for all positive integer powers of [latex]x[\/latex]. We will eventually extend this result to negative integer powers. Later, we will see that this rule may also be extended first to rational powers of [latex]x[\/latex] and then to arbitrary powers of [latex]x[\/latex]. Be aware, however, that this rule does not apply to functions in which a constant is raised to a variable power, such as [latex]f(x)=3^x[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1169736615212\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">The Power Rule<\/h3>\r\n\r\n<hr>\r\n<p id=\"fs-id1169738850005\">Let [latex]n[\/latex] be a positive integer. If [latex]f(x)=x^n[\/latex], then<\/p>\r\n\r\n<div id=\"fs-id1169739269835\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=nx^{n-1}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739031758\">Alternatively, we may express this rule as<\/p>\r\n\r\n<div id=\"fs-id1169739225629\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(x^n)=nx^{n-1}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div id=\"fs-id1169738999189\" class=\"bc-section section\">\r\n<div id=\"fs-id1169739190555\" class=\"textbook exercises\">\r\n<h3>Example: Applying the Power Rule<\/h3>\r\n<p id=\"fs-id1169736613648\">Find the derivative of the function [latex]f(x)=x^{10}[\/latex] by applying the power rule.<\/p>\r\n[reveal-answer q=\"fs-id1169736656146\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736656146\"]\r\n<p id=\"fs-id1169736656146\">Using the power rule with [latex]n=10[\/latex], we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739342007\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=10x^{10-1}=10x^9[\/latex].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736589163\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739199747\">Find the derivative of [latex]f(x)=x^7[\/latex].<\/p>\r\n[reveal-answer q=\"25547709\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"25547709\"]\r\n<p id=\"fs-id1169736613524\">Use the power rule with [latex]n=7[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169738962015\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738962015\"]\r\n<p id=\"fs-id1169738962015\">[latex]f^{\\prime}(x)=7x^6[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1169739270017\">We find our next differentiation rules by looking at derivatives of sums, differences, and constant multiples of functions. Just as when we work with functions, there are rules that make it easier to find derivatives of functions that we add, subtract, or multiply by a constant. These rules are summarized in the following theorem.<\/p>\r\n\r\n<div id=\"fs-id1169739305071\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Sum, Difference, and Constant Multiple Rules<\/h3>\r\n\r\n<hr>\r\n<p id=\"fs-id1169736611426\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be differentiable functions and [latex]k[\/latex] be a constant. Then each of the following equations holds.<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1169739039653\"><strong>Sum Rule:<\/strong>&nbsp;The derivative of the sum of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the sum of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1169739179597\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(f(x)+g(x))=\\frac{d}{dx}(f(x))+\\frac{d}{dx}(g(x))[\/latex];<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739351562\">that is,<\/p>\r\n\r\n<div id=\"fs-id1169739008128\" class=\"equation unnumbered\" style=\"text-align: center;\">for [latex]j(x)=f(x)+g(x), \\, j^{\\prime}(x)=f^{\\prime}(x)+g^{\\prime}(x)[\/latex]<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1169739000178\"><strong>Difference Rule:<\/strong>&nbsp;The derivative of the difference of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the difference of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1169736611311\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(f(x)-g(x))=\\frac{d}{dx}(f(x))-\\frac{d}{dx}(g(x))[\/latex];<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<p id=\"fs-id1169739005930\">that is,<\/p>\r\n\r\n<div id=\"fs-id1169739208882\" class=\"equation unnumbered\" style=\"text-align: center;\">for [latex]j(x)=f(x)-g(x), \\, j^{\\prime}(x)=f^{\\prime}(x)-g^{\\prime}(x)[\/latex]<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1169739195371\"><strong>Constant Multiple Rule:<\/strong>&nbsp;The derivative of a constant [latex]k[\/latex] multiplied by a function [latex]f[\/latex] is the same as the constant multiplied by the derivative:<\/p>\r\n\r\n<div id=\"fs-id1169739340266\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(kf(x))=k\\frac{d}{dx}(f(x))[\/latex];<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739179540\">that is,<\/p>\r\n\r\n<div id=\"fs-id1169739179543\" class=\"equation unnumbered\" style=\"text-align: center;\">for [latex]j(x)=kf(x), \\, j^{\\prime}(x)=kf^{\\prime}(x)[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739300387\" class=\"textbook exercises\">\r\n<h3>Example: Applying Basic Derivative Rules<\/h3>\r\n<p id=\"fs-id1169739300397\">Find the derivative of [latex]f(x)=2x^5+7[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739064774\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739064774\"]\r\n<p id=\"fs-id1169739064774\">We begin by applying the rule for differentiating the sum of two functions, followed by the rules for differentiating constant multiples of functions and the rule for differentiating powers. To better understand the sequence in which the differentiation rules are applied, we use Leibniz notation throughout the solution:<\/p>\r\n\r\n<div id=\"fs-id1169739273990\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}f^{\\prime}(x) &amp; =\\frac{d}{dx}(2x^5+7) &amp; &amp; &amp; \\\\ &amp; =\\frac{d}{dx}(2x^5)+\\frac{d}{dx}(7) &amp; &amp; &amp; \\text{Apply the sum rule.} \\\\ &amp; =2\\frac{d}{dx}(x^5)+\\frac{d}{dx}(7) &amp; &amp; &amp; \\text{Apply the constant multiple rule.} \\\\ &amp; =2(5x^4)+0 &amp; &amp; &amp; \\text{Apply the power rule and the constant rule.} \\\\ &amp; =10x^4. &amp; &amp; &amp; \\text{Simplify.} \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739304168\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739299469\">Find the derivative of [latex]f(x)=2x^3-6x^2+3[\/latex].<\/p>\r\n[reveal-answer q=\"990033\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"990033\"]\r\n<p id=\"fs-id1169739301883\">Use the preceding example as a guide.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169736658726\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736658726\"]\r\n<p id=\"fs-id1169736658726\">[latex]f^{\\prime}(x)=6x^2-12x[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Product Rule<\/h2>\r\n\r\nAlthough it might be tempting to assume that the derivative of the product is the product of the derivatives, similar to the sum and difference rules, the <strong>product rule<\/strong> does not follow this pattern. To see why we cannot use this pattern, consider the function [latex]f(x)=x^2[\/latex], whose derivative is [latex]f^{\\prime}(x)=2x[\/latex] and not [latex]\\frac{d}{dx}(x)\\cdot \\frac{d}{dx}(x)=1\\cdot 1=1[\/latex].<\/p>\r\n<div id=\"fs-id1169739298175\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Product Rule<\/h3>\r\n\r\n<hr>\r\n<p id=\"fs-id1169739298182\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be differentiable functions. Then<\/p>\r\n\r\n<div id=\"fs-id1169739326645\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(f(x)g(x))=\\frac{d}{dx}(f(x))\\cdot g(x)+\\frac{d}{dx}(g(x))\\cdot f(x)[\/latex]<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<p id=\"fs-id1169739187834\">That is,<\/p>\r\n\r\n<div id=\"fs-id1169739187837\" class=\"equation unnumbered\" style=\"text-align: center;\">if [latex]j(x)=f(x)g(x)[\/latex] then [latex]j^{\\prime}(x)=f^{\\prime}(x)g(x)+g^{\\prime}(x)f(x)[\/latex]<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<p id=\"fs-id1169739269717\">This means that the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739273812\" class=\"textbook exercises\">\r\n<h3>Example: Applying the Product Rule to Binomials<\/h3>\r\n<p id=\"fs-id1169739273822\">For [latex]j(x)=(x^2+2)(3x^3-5x)[\/latex], find [latex]j^{\\prime}(x)[\/latex] by applying the product rule.<\/p>\r\n[reveal-answer q=\"fs-id1169739301174\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739301174\"]\r\n<p id=\"fs-id1169739301174\">If we set [latex]f(x)=x^2+2[\/latex] and [latex]g(x)=3x^3-5x[\/latex], then [latex]f^{\\prime}(x)=2x[\/latex] and [latex]g^{\\prime}(x)=9x^2-5[\/latex]. Thus,<\/p>\r\n\r\n<div id=\"fs-id1169739343689\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]j^{\\prime}(x)=f^{\\prime}(x)g(x)+g^{\\prime}(x)f(x)=(2x)(3x^3-5x)+(9x^2-5)(x^2+2)[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169736612557\">Simplifying, we have<\/p>\r\n\r\n<div id=\"fs-id1169736612560\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]j^{\\prime}(x)=15x^4+3x^2-10[\/latex].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736654821\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169736654828\">Use the product rule to obtain the derivative of [latex]j(x)=2x^5(4x^2+x)[\/latex].<\/p>\r\n[reveal-answer q=\"034256\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"034256\"]\r\n<p id=\"fs-id1169736609959\">Set [latex]f(x)=2x^5[\/latex] and [latex]g(x)=4x^2+x[\/latex] and use the preceding example as a guide.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169736654876\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736654876\"]\r\n<p id=\"fs-id1169736654876\">[latex]j^{\\prime}(x)=10x^4(4x^2+x)+(8x+1)(2x^5)=56x^6+12x^5[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Quotient Rule<\/h2>\r\n<p id=\"fs-id1169739269461\">Having developed and practiced the product rule, we now consider differentiating quotients of functions. As we see in the following theorem, the derivative of the quotient is not the quotient of the derivatives; rather, it is the derivative of the function in the numerator times the function in the denominator minus the derivative of the function in the denominator times the function in the numerator, all divided by the square of the function in the denominator. In order to better grasp why we cannot simply take the quotient of the derivatives, keep in mind that<\/p>\r\n\r\n<div id=\"fs-id1169739269470\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(x^2)=2x[\/latex], which is not the same as [latex]\\dfrac{\\frac{d}{dx}(x^3)}{\\frac{d}{dx}(x)} =\\dfrac{3x^2}{1}=3x^2[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<div id=\"fs-id1169736662915\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">The Quotient Rule<\/h3>\r\n\r\n<hr>\r\n<p id=\"fs-id1169736662921\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be differentiable functions. Then<\/p>\r\n\r\n<div id=\"fs-id1169739336009\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}\\left(\\dfrac{f(x)}{g(x)}\\right)=\\dfrac{\\frac{d}{dx}(f(x))\\cdot g(x)-\\dfrac{d}{dx}(g(x))\\cdot f(x)}{(g(x))^2}[\/latex]<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<p id=\"fs-id1169739190663\">That is,<\/p>\r\n\r\n<div id=\"fs-id1169739190666\" class=\"equation unnumbered\" style=\"text-align: center;\">if [latex]j(x)=\\dfrac{f(x)}{g(x)}[\/latex], then [latex]j^{\\prime}(x)=\\dfrac{f^{\\prime}(x)g(x)-g^{\\prime}(x)f(x)}{(g(x))^2}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739305225\" class=\"textbook exercises\">\r\n<h3>Example: Applying the Quotient Rule<\/h3>\r\n<p id=\"fs-id1169739305235\">Use the quotient rule to find the derivative of [latex]k(x)=\\dfrac{5x^2}{4x+3}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739305276\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739305276\"]\r\n<p id=\"fs-id1169739305276\">Let [latex]f(x)=5x^2[\/latex] and [latex]g(x)=4x+3[\/latex]. Thus, [latex]f^{\\prime}(x)=10x[\/latex] and [latex]g^{\\prime}(x)=4[\/latex]. Substituting into the quotient rule, we have<\/p>\r\n\r\n<div id=\"fs-id1169739299200\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]k^{\\prime}(x)=\\dfrac{f^{\\prime}(x)g(x)-g^{\\prime}(x)f(x)}{(g(x))^2}=\\dfrac{10x(4x+3)-4(5x^2)}{(4x+3)^2}[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739299784\">Simplifying, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739299787\" class=\"equation unnumbered\">[latex]k^{\\prime}(x)=\\dfrac{20x^2+30x}{(4x+3)^2}[\/latex].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739299850\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739299858\">Find the derivative of [latex]h(x)=\\dfrac{3x+1}{4x-3}[\/latex]<\/p>\r\n[reveal-answer q=\"336671\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"336671\"]\r\n<p id=\"fs-id1169739348450\">Apply the quotient rule with [latex]f(x)=3x+1[\/latex] and [latex]g(x)=4x-3[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169739348394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739348394\"]\r\n<p id=\"fs-id1169739348394\">[latex]k^{\\prime}(x)=-\\dfrac{13}{(4x-3)^2}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>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.<\/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>Indefinite Integrals<\/h2>\r\n<div id=\"fs-id1165043041347\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Power Rule for Integrals<\/h3>\r\n\r\n<hr>\r\n<p id=\"fs-id1165042514785\">For [latex]n \\ne \u22121[\/latex],<\/p>\r\n\r\n<div id=\"fs-id1165043250161\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int x^n dx=\\dfrac{x^{n+1}}{n+1}+C[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<p id=\"fs-id1165043385541\">Evaluating indefinite integrals for some other functions is also a straightforward calculation. The following table lists the indefinite integrals for several common functions. A more complete list appears in <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/appendix-b-table-of-derivatives\/\" target=\"_blank\" rel=\"noopener\">Appendix B: Table of Derivatives<\/a>.<\/p>\r\n\r\n<table summary=\"This is a table with two columns and fourteen rows, titled \u201cIntegration Formulas.\u201d The first row is a header row, and labels column one \u201cDifferentiation Formula\u201d and column two \u201cIndefinite Integral.\u201d The second row reads d\/dx (k) = 0, the integral of kdx = the integral of kx^0dx = kx + C. The third row reads d\/dx(x^n) = nx^(x-1), the integral of x^ndn = (x^n+1)\/(n+1) + C for n is not equal to negative 1. The fourth row reads d\/dx(ln(the absolute value of x))=1\/x, the integral of (1\/x)dx = ln(the absolute value of x) + C. The fifth row reads d\/dx(e^x) = e^x, the integral of e^xdx = e^x + C. The sixth row reads d\/dx(sinx) = cosx, the integral of cosxdx = sinx + C. The seventh row reads d\/dx(cosx) = negative sinx, the integral of sinxdx = negative cosx + C. The eighth row reads d\/dx(tanx) = sec squared x, the integral of sec squared xdx = tanx + C. The ninth row reads d\/dx(cscx) = negative cscxcotx, the integral of cscxcotxdx = negative cscx + C. The tenth row reads d\/dx(secx) = secxtanx, the integral of secxtanxdx = secx + C. The eleventh row reads d\/dx(cotx) = negative csc squared x, the integral of csc squared xdx = negative cot x + C. The twelfth row reads d\/dx(sin^-1(x)) = 1\/the square root of (1 \u2013 x^2), the integral of 1\/(the square root of (x^2 \u2013 1) = sin^-1(x) + C. The thirteenth row reads d\/dx (tan^-1(x)) = 1\/(1 + x^2), the integral of 1\/(1 + x^2)dx = tan^-1(x) + C. The fourteenth row reads d\/dx(sec^-1(the absolute value of x)) = 1\/x(the square root of x^2 \u2013 1), the integral of 1\/x(the square root of x^2 \u2013 1)dx = sec^-1(the absolute value of x) + C.\"><caption>Integration Formulas<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Differentiation Formula<\/th>\r\n<th>Indefinite Integral<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(k)=0[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int kdx=\\displaystyle\\int kx^0 dx=kx+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(x^n)=nx^{n-1}[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int x^n dx=\\frac{x^{n+1}}{n+1}+C[\/latex] for [latex]n\\ne \u22121[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\ln |x|)=\\frac{1}{x}[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int \\frac{1}{x}dx=\\ln |x|+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(e^x)=e^x[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int e^x dx=e^x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int \\cos x dx= \\sin x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\cos x)=\u2212 \\sin x[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int \\sin x dx=\u2212 \\cos x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\tan x)= \\sec^2 x[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int \\sec^2 x dx= \\tan x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int \\csc x \\cot x dx=\u2212\\csc x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int \\sec x \\tan x dx= \\sec x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int \\csc^2 x dx=\u2212\\cot x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}( \\sin^{-1} x)=\\frac{1}{\\sqrt{1-x^2}}[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int \\frac{1}{\\sqrt{1-x^2}} dx= \\sin^{-1} x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\tan^{-1} x)=\\frac{1}{1+x^2}[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int \\frac{1}{1+x^2} dx= \\tan^{-1} x+C[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{d}{dx}(\\sec^{-1} |x|)=\\frac{1}{x\\sqrt{x^2-1}}[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\int \\frac{1}{x\\sqrt{x^2-1}} dx= \\sec^{-1} |x|+C[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-id1165043248811\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating Indefinite Integrals<\/h3>\r\nEvaluate each of the following indefinite integrals:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\displaystyle\\int (5x^3-7x^2+3x+4) dx[\/latex]<\/li>\r\n \t<li>[latex]\\displaystyle\\int \\frac{x^2+4\\sqrt[3]{x}}{x} dx[\/latex]<\/li>\r\n \t<li>[latex]\\displaystyle\\int \\frac{4}{1+x^2} dx[\/latex]<\/li>\r\n \t<li>[latex]\\displaystyle\\int \\tan x \\cos x dx[\/latex]<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1165042705917\" class=\"exercise\">[reveal-answer q=\"fs-id1165042552215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042552215\"]\r\n<ol id=\"fs-id1165042552215\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>Using the properties of indefinite integrals, we can integrate each of the four terms in the integrand separately. We obtain\r\n<div id=\"fs-id1165042552227\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int (5x^3-7x^2+3x+4) dx=\\displaystyle\\int 5x^3 dx-\\displaystyle\\int 7x^2 dx+\\displaystyle\\int 3x dx+\\displaystyle\\int 4 dx[\/latex]<\/div>\r\nFrom the Constant Multiples property of indefinite integrals, each coefficient can be written in front of the integral sign, which gives\r\n<div id=\"fs-id1165043312575\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int 5x^3 dx-\\displaystyle\\int 7x^2 dx+\\displaystyle\\int 3x dx+\\displaystyle\\int 4 dx=5\\displaystyle\\int x^3 dx-7\\displaystyle\\int x^2 dx+3\\displaystyle\\int x dx+4\\displaystyle\\int 1 dx[\/latex]<\/div>\r\nUsing the power rule for integrals, we conclude that\r\n<div id=\"fs-id1165042407363\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int (5x^3-7x^2+3x+4) dx=\\frac{5}{4}x^4-\\frac{7}{3}x^3+\\frac{3}{2}x^2+4x+C[\/latex]<\/div><\/li>\r\n \t<li>Rewrite the integrand as\r\n<div id=\"fs-id1165042371846\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{x^2+4\\sqrt[3]{x}}{x}=\\frac{x^2}{x}+\\frac{4\\sqrt[3]{x}}{x}[\/latex]<\/div>\r\nThen, to evaluate the integral, integrate each of these terms separately. Using the power rule, we have\r\n<div id=\"fs-id1165043427498\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll} \\displaystyle\\int (x+\\frac{4}{x^{2\/3}}) dx &amp; =\\displaystyle\\int x dx+4\\displaystyle\\int x^{-2\/3} dx \\\\ &amp; =\\frac{1}{2}x^2+4\\frac{1}{(\\frac{-2}{3})+1}x^{(-2\/3)+1}+C \\\\ &amp; =\\frac{1}{2}x^2+12x^{1\/3}+C \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Using the properties of indefinite integrals, write the integral as\r\n<div id=\"fs-id1165043348665\" class=\"equation unnumbered\">[latex]4\\displaystyle\\int \\frac{1}{1+x^2} dx[\/latex].<\/div>\r\nThen, use the fact that [latex] \\tan^{-1} (x)[\/latex] is an antiderivative of [latex]\\frac{1}{1+x^2}[\/latex] to conclude that\r\n<div id=\"fs-id1165042374764\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int \\frac{4}{1+x^2} dx=4 \\tan^{-1} (x)+C[\/latex]<\/div><\/li>\r\n \t<li>Rewrite the integrand as\r\n<div class=\"equation unnumbered\">[latex] \\tan x \\cos x=\\frac{ \\sin x}{ \\cos x} \\cos x= \\sin x[\/latex].<\/div>\r\nTherefore,\r\n<div id=\"fs-id1165043317182\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int \\tan x \\cos x dx=\\displaystyle\\int \\sin x dx=\u2212 \\cos x+C[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nEvaluate [latex]\\displaystyle\\int (4x^3-5x^2+x-7) dx[\/latex]\r\n\r\n[reveal-answer q=\"4078823\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"4078823\"]\r\n\r\nIntegrate each term in the integrand separately, making use of the power rule.\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1165043259694\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043259694\"]\r\n\r\n[latex]x^4-\\frac{5}{3}x^3+\\frac{1}{2}x^2-7x+C[\/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\n[ohm_question]210143[\/ohm_question]\r\n\r\n<\/div>\r\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Apply basic derivative rules<\/li>\n<li>Use the product rule for finding the derivative of a product of functions<\/li>\n<li>Use the quotient rule for finding the derivative of a quotient of functions<\/li>\n<li>Apply the chain rule together with the power and product rule<\/li>\n<li>Evaluate indefinite integrals<\/li>\n<\/ul>\n<\/div>\n<p>In the Calculus of Vector-Valued Functions section, we will learn how to differentiate and integrate vector-valued functions. Here we will review various derivative rules and integration techniques.<\/p>\n<h2>Basic Derivative Rules<\/h2>\n<p id=\"fs-id1169738835554\">We first apply the limit definition of the derivative to find the derivative of the constant function, [latex]f(x)=c[\/latex]. For this function, both [latex]f(x)=c[\/latex] and [latex]f(x+h)=c[\/latex], so we obtain the following result:<\/p>\n<div id=\"fs-id1169738850732\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}f^{\\prime}(x) & =\\underset{h\\to 0}{\\lim}\\dfrac{f(x+h)-f(x)}{h} \\\\ & =\\underset{h\\to 0}{\\lim}\\dfrac{c-c}{h} \\\\ & =\\underset{h\\to 0}{\\lim}\\dfrac{0}{h} \\\\ & =\\underset{h\\to 0}{\\lim}0=0 \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738850205\">The rule for differentiating constant functions is called the <strong>constant rule<\/strong>. It states that the derivative of a constant function is zero; that is, since a constant function is a horizontal line, the slope, or the rate of change, of a constant function is 0. We restate this rule in the following theorem.<\/p>\n<div id=\"fs-id1169738828810\" class=\"bc-section section\">\n<div id=\"fs-id1169739030384\" class=\"bc-section section\">\n<div id=\"fs-id1169738955425\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">The Constant Rule<\/h3>\n<hr \/>\n<p id=\"fs-id1169738878658\">Let [latex]c[\/latex] be a constant.<\/p>\n<p id=\"fs-id1169738853363\" style=\"text-align: center;\">If [latex]f(x)=c[\/latex], then [latex]f^{\\prime}(c)=0[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739024163\">Alternatively, we may express this rule as<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(c)=0[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1169739274547\" class=\"textbook exercises\">\n<h3>Example: Applying the Constant Rule<\/h3>\n<p id=\"fs-id1169738824417\">Find the derivative of [latex]f(x)=8[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738865666\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738865666\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738865666\">This is just a one-step application of the rule:<\/p>\n<div id=\"fs-id1169738875468\" class=\"equation unnumbered\">[latex]f^{\\prime}(8)=0[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738954922\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169736614166\">Find the derivative of [latex]g(x)=-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738853102\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738853102\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738853102\">0<\/p>\n<h4>Hint<\/h4>\n<p id=\"fs-id1169738907507\">Use the preceding example as a guide.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169739006200\">We have shown that<\/p>\n<div id=\"fs-id1169739234394\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\dfrac{d}{dx}\\left(x^2\\right)=2x[\/latex]\u00a0 \u00a0and\u00a0 \u00a0[latex]\\dfrac{d}{dx}\\left(x^{\\frac{1}{2}}\\right)=\\dfrac{1}{2}x^{\u2212\\frac{1}{2}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738969429\">At this point, you might see a pattern beginning to develop for derivatives of the form [latex]\\frac{d}{dx}(x^n)[\/latex]. We continue our examination of derivative formulas by differentiating power functions of the form [latex]f(x)=x^n[\/latex] where [latex]n[\/latex] is a positive integer. We develop formulas for derivatives of this type of function in stages, beginning with positive integer powers. Before stating and proving the general rule for derivatives of functions of this form, we take a look at a specific case, [latex]\\frac{d}{dx}(x^3)[\/latex].<\/p>\n<p id=\"fs-id1169736619689\">As we shall see, the procedure for finding the derivative of the general form [latex]f(x)=x^n[\/latex] is very similar. Although it is often unwise to draw general conclusions from specific examples, we note that when we differentiate [latex]f(x)=x^3[\/latex], the power on [latex]x[\/latex] becomes the coefficient of [latex]x^2[\/latex] in the derivative and the power on [latex]x[\/latex] in the derivative decreases by 1. The following theorem states that this\u00a0<strong>power rule<\/strong> holds for all positive integer powers of [latex]x[\/latex]. We will eventually extend this result to negative integer powers. Later, we will see that this rule may also be extended first to rational powers of [latex]x[\/latex] and then to arbitrary powers of [latex]x[\/latex]. Be aware, however, that this rule does not apply to functions in which a constant is raised to a variable power, such as [latex]f(x)=3^x[\/latex].<\/p>\n<div id=\"fs-id1169736615212\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">The Power Rule<\/h3>\n<hr \/>\n<p id=\"fs-id1169738850005\">Let [latex]n[\/latex] be a positive integer. If [latex]f(x)=x^n[\/latex], then<\/p>\n<div id=\"fs-id1169739269835\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=nx^{n-1}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739031758\">Alternatively, we may express this rule as<\/p>\n<div id=\"fs-id1169739225629\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(x^n)=nx^{n-1}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1169738999189\" class=\"bc-section section\">\n<div id=\"fs-id1169739190555\" class=\"textbook exercises\">\n<h3>Example: Applying the Power Rule<\/h3>\n<p id=\"fs-id1169736613648\">Find the derivative of the function [latex]f(x)=x^{10}[\/latex] by applying the power rule.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736656146\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736656146\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736656146\">Using the power rule with [latex]n=10[\/latex], we obtain<\/p>\n<div id=\"fs-id1169739342007\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=10x^{10-1}=10x^9[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736589163\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739199747\">Find the derivative of [latex]f(x)=x^7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q25547709\">Hint<\/span><\/p>\n<div id=\"q25547709\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736613524\">Use the power rule with [latex]n=7[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738962015\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738962015\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738962015\">[latex]f^{\\prime}(x)=7x^6[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169739270017\">We find our next differentiation rules by looking at derivatives of sums, differences, and constant multiples of functions. Just as when we work with functions, there are rules that make it easier to find derivatives of functions that we add, subtract, or multiply by a constant. These rules are summarized in the following theorem.<\/p>\n<div id=\"fs-id1169739305071\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Sum, Difference, and Constant Multiple Rules<\/h3>\n<hr \/>\n<p id=\"fs-id1169736611426\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be differentiable functions and [latex]k[\/latex] be a constant. Then each of the following equations holds.<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739039653\"><strong>Sum Rule:<\/strong>&nbsp;The derivative of the sum of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the sum of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex].<\/p>\n<div id=\"fs-id1169739179597\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(f(x)+g(x))=\\frac{d}{dx}(f(x))+\\frac{d}{dx}(g(x))[\/latex];<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739351562\">that is,<\/p>\n<div id=\"fs-id1169739008128\" class=\"equation unnumbered\" style=\"text-align: center;\">for [latex]j(x)=f(x)+g(x), \\, j^{\\prime}(x)=f^{\\prime}(x)+g^{\\prime}(x)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<div><\/div>\n<p id=\"fs-id1169739000178\"><strong>Difference Rule:<\/strong>&nbsp;The derivative of the difference of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the difference of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex].<\/p>\n<div id=\"fs-id1169736611311\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(f(x)-g(x))=\\frac{d}{dx}(f(x))-\\frac{d}{dx}(g(x))[\/latex];<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<p id=\"fs-id1169739005930\">that is,<\/p>\n<div id=\"fs-id1169739208882\" class=\"equation unnumbered\" style=\"text-align: center;\">for [latex]j(x)=f(x)-g(x), \\, j^{\\prime}(x)=f^{\\prime}(x)-g^{\\prime}(x)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<div><\/div>\n<p id=\"fs-id1169739195371\"><strong>Constant Multiple Rule:<\/strong>&nbsp;The derivative of a constant [latex]k[\/latex] multiplied by a function [latex]f[\/latex] is the same as the constant multiplied by the derivative:<\/p>\n<div id=\"fs-id1169739340266\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(kf(x))=k\\frac{d}{dx}(f(x))[\/latex];<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739179540\">that is,<\/p>\n<div id=\"fs-id1169739179543\" class=\"equation unnumbered\" style=\"text-align: center;\">for [latex]j(x)=kf(x), \\, j^{\\prime}(x)=kf^{\\prime}(x)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1169739300387\" class=\"textbook exercises\">\n<h3>Example: Applying Basic Derivative Rules<\/h3>\n<p id=\"fs-id1169739300397\">Find the derivative of [latex]f(x)=2x^5+7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739064774\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739064774\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739064774\">We begin by applying the rule for differentiating the sum of two functions, followed by the rules for differentiating constant multiples of functions and the rule for differentiating powers. To better understand the sequence in which the differentiation rules are applied, we use Leibniz notation throughout the solution:<\/p>\n<div id=\"fs-id1169739273990\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}f^{\\prime}(x) & =\\frac{d}{dx}(2x^5+7) & & & \\\\ & =\\frac{d}{dx}(2x^5)+\\frac{d}{dx}(7) & & & \\text{Apply the sum rule.} \\\\ & =2\\frac{d}{dx}(x^5)+\\frac{d}{dx}(7) & & & \\text{Apply the constant multiple rule.} \\\\ & =2(5x^4)+0 & & & \\text{Apply the power rule and the constant rule.} \\\\ & =10x^4. & & & \\text{Simplify.} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739304168\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739299469\">Find the derivative of [latex]f(x)=2x^3-6x^2+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q990033\">Hint<\/span><\/p>\n<div id=\"q990033\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739301883\">Use the preceding example as a guide.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736658726\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736658726\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736658726\">[latex]f^{\\prime}(x)=6x^2-12x[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>The Product Rule<\/h2>\n<p>Although it might be tempting to assume that the derivative of the product is the product of the derivatives, similar to the sum and difference rules, the <strong>product rule<\/strong> does not follow this pattern. To see why we cannot use this pattern, consider the function [latex]f(x)=x^2[\/latex], whose derivative is [latex]f^{\\prime}(x)=2x[\/latex] and not [latex]\\frac{d}{dx}(x)\\cdot \\frac{d}{dx}(x)=1\\cdot 1=1[\/latex].<\/p>\n<div id=\"fs-id1169739298175\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Product Rule<\/h3>\n<hr \/>\n<p id=\"fs-id1169739298182\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be differentiable functions. Then<\/p>\n<div id=\"fs-id1169739326645\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(f(x)g(x))=\\frac{d}{dx}(f(x))\\cdot g(x)+\\frac{d}{dx}(g(x))\\cdot f(x)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<p id=\"fs-id1169739187834\">That is,<\/p>\n<div id=\"fs-id1169739187837\" class=\"equation unnumbered\" style=\"text-align: center;\">if [latex]j(x)=f(x)g(x)[\/latex] then [latex]j^{\\prime}(x)=f^{\\prime}(x)g(x)+g^{\\prime}(x)f(x)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<p id=\"fs-id1169739269717\">This means that the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.<\/p>\n<\/div>\n<div id=\"fs-id1169739273812\" class=\"textbook exercises\">\n<h3>Example: Applying the Product Rule to Binomials<\/h3>\n<p id=\"fs-id1169739273822\">For [latex]j(x)=(x^2+2)(3x^3-5x)[\/latex], find [latex]j^{\\prime}(x)[\/latex] by applying the product rule.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739301174\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739301174\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739301174\">If we set [latex]f(x)=x^2+2[\/latex] and [latex]g(x)=3x^3-5x[\/latex], then [latex]f^{\\prime}(x)=2x[\/latex] and [latex]g^{\\prime}(x)=9x^2-5[\/latex]. Thus,<\/p>\n<div id=\"fs-id1169739343689\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]j^{\\prime}(x)=f^{\\prime}(x)g(x)+g^{\\prime}(x)f(x)=(2x)(3x^3-5x)+(9x^2-5)(x^2+2)[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169736612557\">Simplifying, we have<\/p>\n<div id=\"fs-id1169736612560\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]j^{\\prime}(x)=15x^4+3x^2-10[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736654821\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169736654828\">Use the product rule to obtain the derivative of [latex]j(x)=2x^5(4x^2+x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q034256\">Hint<\/span><\/p>\n<div id=\"q034256\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736609959\">Set [latex]f(x)=2x^5[\/latex] and [latex]g(x)=4x^2+x[\/latex] and use the preceding example as a guide.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736654876\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736654876\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736654876\">[latex]j^{\\prime}(x)=10x^4(4x^2+x)+(8x+1)(2x^5)=56x^6+12x^5[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>The Quotient Rule<\/h2>\n<p id=\"fs-id1169739269461\">Having developed and practiced the product rule, we now consider differentiating quotients of functions. As we see in the following theorem, the derivative of the quotient is not the quotient of the derivatives; rather, it is the derivative of the function in the numerator times the function in the denominator minus the derivative of the function in the denominator times the function in the numerator, all divided by the square of the function in the denominator. In order to better grasp why we cannot simply take the quotient of the derivatives, keep in mind that<\/p>\n<div id=\"fs-id1169739269470\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(x^2)=2x[\/latex], which is not the same as [latex]\\dfrac{\\frac{d}{dx}(x^3)}{\\frac{d}{dx}(x)} =\\dfrac{3x^2}{1}=3x^2[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div id=\"fs-id1169736662915\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">The Quotient Rule<\/h3>\n<hr \/>\n<p id=\"fs-id1169736662921\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be differentiable functions. Then<\/p>\n<div id=\"fs-id1169739336009\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}\\left(\\dfrac{f(x)}{g(x)}\\right)=\\dfrac{\\frac{d}{dx}(f(x))\\cdot g(x)-\\dfrac{d}{dx}(g(x))\\cdot f(x)}{(g(x))^2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<p id=\"fs-id1169739190663\">That is,<\/p>\n<div id=\"fs-id1169739190666\" class=\"equation unnumbered\" style=\"text-align: center;\">if [latex]j(x)=\\dfrac{f(x)}{g(x)}[\/latex], then [latex]j^{\\prime}(x)=\\dfrac{f^{\\prime}(x)g(x)-g^{\\prime}(x)f(x)}{(g(x))^2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1169739305225\" class=\"textbook exercises\">\n<h3>Example: Applying the Quotient Rule<\/h3>\n<p id=\"fs-id1169739305235\">Use the quotient rule to find the derivative of [latex]k(x)=\\dfrac{5x^2}{4x+3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739305276\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739305276\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739305276\">Let [latex]f(x)=5x^2[\/latex] and [latex]g(x)=4x+3[\/latex]. Thus, [latex]f^{\\prime}(x)=10x[\/latex] and [latex]g^{\\prime}(x)=4[\/latex]. Substituting into the quotient rule, we have<\/p>\n<div id=\"fs-id1169739299200\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]k^{\\prime}(x)=\\dfrac{f^{\\prime}(x)g(x)-g^{\\prime}(x)f(x)}{(g(x))^2}=\\dfrac{10x(4x+3)-4(5x^2)}{(4x+3)^2}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739299784\">Simplifying, we obtain<\/p>\n<div id=\"fs-id1169739299787\" class=\"equation unnumbered\">[latex]k^{\\prime}(x)=\\dfrac{20x^2+30x}{(4x+3)^2}[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739299850\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739299858\">Find the derivative of [latex]h(x)=\\dfrac{3x+1}{4x-3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q336671\">Hint<\/span><\/p>\n<div id=\"q336671\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739348450\">Apply the quotient rule with [latex]f(x)=3x+1[\/latex] and [latex]g(x)=4x-3[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739348394\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739348394\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739348394\">[latex]k^{\\prime}(x)=-\\dfrac{13}{(4x-3)^2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>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.<\/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>Indefinite Integrals<\/h2>\n<div id=\"fs-id1165043041347\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Power Rule for Integrals<\/h3>\n<hr \/>\n<p id=\"fs-id1165042514785\">For [latex]n \\ne \u22121[\/latex],<\/p>\n<div id=\"fs-id1165043250161\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int x^n dx=\\dfrac{x^{n+1}}{n+1}+C[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p id=\"fs-id1165043385541\">Evaluating indefinite integrals for some other functions is also a straightforward calculation. The following table lists the indefinite integrals for several common functions. A more complete list appears in <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/appendix-b-table-of-derivatives\/\" target=\"_blank\" rel=\"noopener\">Appendix B: Table of Derivatives<\/a>.<\/p>\n<table summary=\"This is a table with two columns and fourteen rows, titled \u201cIntegration Formulas.\u201d The first row is a header row, and labels column one \u201cDifferentiation Formula\u201d and column two \u201cIndefinite Integral.\u201d The second row reads d\/dx (k) = 0, the integral of kdx = the integral of kx^0dx = kx + C. The third row reads d\/dx(x^n) = nx^(x-1), the integral of x^ndn = (x^n+1)\/(n+1) + C for n is not equal to negative 1. The fourth row reads d\/dx(ln(the absolute value of x))=1\/x, the integral of (1\/x)dx = ln(the absolute value of x) + C. The fifth row reads d\/dx(e^x) = e^x, the integral of e^xdx = e^x + C. The sixth row reads d\/dx(sinx) = cosx, the integral of cosxdx = sinx + C. The seventh row reads d\/dx(cosx) = negative sinx, the integral of sinxdx = negative cosx + C. The eighth row reads d\/dx(tanx) = sec squared x, the integral of sec squared xdx = tanx + C. The ninth row reads d\/dx(cscx) = negative cscxcotx, the integral of cscxcotxdx = negative cscx + C. The tenth row reads d\/dx(secx) = secxtanx, the integral of secxtanxdx = secx + C. The eleventh row reads d\/dx(cotx) = negative csc squared x, the integral of csc squared xdx = negative cot x + C. The twelfth row reads d\/dx(sin^-1(x)) = 1\/the square root of (1 \u2013 x^2), the integral of 1\/(the square root of (x^2 \u2013 1) = sin^-1(x) + C. The thirteenth row reads d\/dx (tan^-1(x)) = 1\/(1 + x^2), the integral of 1\/(1 + x^2)dx = tan^-1(x) + C. The fourteenth row reads d\/dx(sec^-1(the absolute value of x)) = 1\/x(the square root of x^2 \u2013 1), the integral of 1\/x(the square root of x^2 \u2013 1)dx = sec^-1(the absolute value of x) + C.\">\n<caption>Integration Formulas<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>Differentiation Formula<\/th>\n<th>Indefinite Integral<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(k)=0[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int kdx=\\displaystyle\\int kx^0 dx=kx+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(x^n)=nx^{n-1}[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int x^n dx=\\frac{x^{n+1}}{n+1}+C[\/latex] for [latex]n\\ne \u22121[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\ln |x|)=\\frac{1}{x}[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int \\frac{1}{x}dx=\\ln |x|+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(e^x)=e^x[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int e^x dx=e^x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int \\cos x dx= \\sin x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\cos x)=\u2212 \\sin x[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int \\sin x dx=\u2212 \\cos x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\tan x)= \\sec^2 x[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int \\sec^2 x dx= \\tan x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int \\csc x \\cot x dx=\u2212\\csc x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int \\sec x \\tan x dx= \\sec x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int \\csc^2 x dx=\u2212\\cot x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}( \\sin^{-1} x)=\\frac{1}{\\sqrt{1-x^2}}[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int \\frac{1}{\\sqrt{1-x^2}} dx= \\sin^{-1} x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\tan^{-1} x)=\\frac{1}{1+x^2}[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int \\frac{1}{1+x^2} dx= \\tan^{-1} x+C[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{d}{dx}(\\sec^{-1} |x|)=\\frac{1}{x\\sqrt{x^2-1}}[\/latex]<\/td>\n<td>[latex]\\displaystyle\\int \\frac{1}{x\\sqrt{x^2-1}} dx= \\sec^{-1} |x|+C[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165043248811\" class=\"textbook exercises\">\n<h3>Example: Evaluating Indefinite Integrals<\/h3>\n<p>Evaluate each of the following indefinite integrals:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\displaystyle\\int (5x^3-7x^2+3x+4) dx[\/latex]<\/li>\n<li>[latex]\\displaystyle\\int \\frac{x^2+4\\sqrt[3]{x}}{x} dx[\/latex]<\/li>\n<li>[latex]\\displaystyle\\int \\frac{4}{1+x^2} dx[\/latex]<\/li>\n<li>[latex]\\displaystyle\\int \\tan x \\cos x dx[\/latex]<\/li>\n<\/ol>\n<div id=\"fs-id1165042705917\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042552215\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042552215\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165042552215\" style=\"list-style-type: lower-alpha;\">\n<li>Using the properties of indefinite integrals, we can integrate each of the four terms in the integrand separately. We obtain\n<div id=\"fs-id1165042552227\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int (5x^3-7x^2+3x+4) dx=\\displaystyle\\int 5x^3 dx-\\displaystyle\\int 7x^2 dx+\\displaystyle\\int 3x dx+\\displaystyle\\int 4 dx[\/latex]<\/div>\n<p>From the Constant Multiples property of indefinite integrals, each coefficient can be written in front of the integral sign, which gives<\/p>\n<div id=\"fs-id1165043312575\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int 5x^3 dx-\\displaystyle\\int 7x^2 dx+\\displaystyle\\int 3x dx+\\displaystyle\\int 4 dx=5\\displaystyle\\int x^3 dx-7\\displaystyle\\int x^2 dx+3\\displaystyle\\int x dx+4\\displaystyle\\int 1 dx[\/latex]<\/div>\n<p>Using the power rule for integrals, we conclude that<\/p>\n<div id=\"fs-id1165042407363\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int (5x^3-7x^2+3x+4) dx=\\frac{5}{4}x^4-\\frac{7}{3}x^3+\\frac{3}{2}x^2+4x+C[\/latex]<\/div>\n<\/li>\n<li>Rewrite the integrand as\n<div id=\"fs-id1165042371846\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{x^2+4\\sqrt[3]{x}}{x}=\\frac{x^2}{x}+\\frac{4\\sqrt[3]{x}}{x}[\/latex]<\/div>\n<p>Then, to evaluate the integral, integrate each of these terms separately. Using the power rule, we have<\/p>\n<div id=\"fs-id1165043427498\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll} \\displaystyle\\int (x+\\frac{4}{x^{2\/3}}) dx & =\\displaystyle\\int x dx+4\\displaystyle\\int x^{-2\/3} dx \\\\ & =\\frac{1}{2}x^2+4\\frac{1}{(\\frac{-2}{3})+1}x^{(-2\/3)+1}+C \\\\ & =\\frac{1}{2}x^2+12x^{1\/3}+C \\end{array}[\/latex]<\/div>\n<\/li>\n<li>Using the properties of indefinite integrals, write the integral as\n<div id=\"fs-id1165043348665\" class=\"equation unnumbered\">[latex]4\\displaystyle\\int \\frac{1}{1+x^2} dx[\/latex].<\/div>\n<p>Then, use the fact that [latex]\\tan^{-1} (x)[\/latex] is an antiderivative of [latex]\\frac{1}{1+x^2}[\/latex] to conclude that<\/p>\n<div id=\"fs-id1165042374764\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int \\frac{4}{1+x^2} dx=4 \\tan^{-1} (x)+C[\/latex]<\/div>\n<\/li>\n<li>Rewrite the integrand as\n<div class=\"equation unnumbered\">[latex]\\tan x \\cos x=\\frac{ \\sin x}{ \\cos x} \\cos x= \\sin x[\/latex].<\/div>\n<p>Therefore,<\/p>\n<div id=\"fs-id1165043317182\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int \\tan x \\cos x dx=\\displaystyle\\int \\sin x dx=\u2212 \\cos x+C[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Evaluate [latex]\\displaystyle\\int (4x^3-5x^2+x-7) dx[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q4078823\">Hint<\/span><\/p>\n<div id=\"q4078823\" class=\"hidden-answer\" style=\"display: none\">\n<p>Integrate each term in the integrand separately, making use of the power rule.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043259694\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043259694\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x^4-\\frac{5}{3}x^3+\\frac{1}{2}x^2-7x+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm210143\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=210143&theme=oea&iframe_resize_id=ohm210143&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n","protected":false},"author":349141,"menu_order":2,"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-4090","chapter","type-chapter","status-publish","hentry"],"part":4118,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4090","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4090\/revisions"}],"predecessor-version":[{"id":6094,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4090\/revisions\/6094"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/4118"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4090\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=4090"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=4090"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=4090"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=4090"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}