{"id":1983,"date":"2021-08-19T16:06:59","date_gmt":"2021-08-19T16:06:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-integration-by-parts\/"},"modified":"2021-11-19T02:59:44","modified_gmt":"2021-11-19T02:59:44","slug":"skills-review-for-integration-by-parts","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-integration-by-parts\/","title":{"raw":"Skills Review for Integration by Parts","rendered":"Skills Review for Integration by Parts"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Apply the power rule<\/li>\r\n \t<li>Find the derivatives of the sine and cosine function.<\/li>\r\n \t<li>Find the derivatives of the standard trigonometric functions.<\/li>\r\n \t<li>Find the derivative of exponential functions<\/li>\r\n \t<li>Find the derivative of logarithmic functions<\/li>\r\n \t<li>Apply substitution integration shortcut formulas<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Integration by Parts section, we will learn how to evaluate integrals where one part of the integral is easily differentiable while the other part is easily integrable. Here we will review some derivative-taking techniques along with substitution integration shortcuts.\r\n<h2>Apply the Power Rule<\/h2>\r\n<p id=\"fs-id1169739006200\">We know 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<p id=\"fs-id1169736619689\">As we shall see, there is a procedure for finding the derivative of the general form [latex]f(x)=x^n[\/latex]. The following theorem states that this\u00a0<strong>power rule<\/strong> holds for all non-variable powers of [latex]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 number. 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<\/div>\r\n<div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Applying Basic Derivative Rules<\/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 class=\"textbox 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{Take the derivative term by term.} \\\\ &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 class=\"textbox exercises\">\r\n<h3>Example: Applying Basic Derivative Rules<\/h3>\r\n<p id=\"fs-id1169739300397\">Find the derivative of [latex]f(x)=\\sqrt{x}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739064780\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739064780\"]\r\n<p id=\"fs-id1169739064774\">For any function in radical form, rewrite it in exponential form.<\/p>\r\n[latex]f(x)=\\sqrt{x}[\/latex] can be rewritten as [latex]f(x)=x^\\frac{1}{2}[\/latex].\r\n\r\nNow, take the derivative using the power rule: [latex]f^{\\prime}(x)=\\dfrac{1}{2}x^\\frac{-1}{2}[\/latex].\r\n<div id=\"fs-id1169739273990\" class=\"equation unnumbered\" style=\"text-align: center;\"><\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox 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=\"fs-id1169736658726\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736658726\"]\r\n<p id=\"fs-id1169736658726\">[latex]f^{\\prime}(x)=-6x^{-4}-12x[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739299469\">Find the derivative of [latex]f(x)=\\sqrt{x^7}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169736658740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736658740\"]\r\n<p id=\"fs-id1169736658726\">Rewrite:[latex]f(x)=x^\\frac{7}{2}[\/latex]<\/p>\r\nDifferentiate: [latex]f^{\\prime}(x)=\\dfrac{7}{2}x^\\frac{5}{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<h2>Find the Derivatives of the Standard Trigonometric Functions.<\/h2>\r\n<div id=\"fs-id1169739098813\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">The Derivatives of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex]<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1169738998734\">The derivative of the sine function is the cosine and the derivative of the cosine function is the negative sine.<\/p>\r\n\r\n<div id=\"fs-id1169738884040\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<div class=\"equation\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex]<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Differentiating a Function Containing [latex]sinx[\/latex]<\/h3>\r\n<p id=\"fs-id1169739269454\">Find the derivative of [latex]f(x)=5x^3 \\sin x[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739028319\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739028319\"]Using the product rule, we have\r\n<div id=\"fs-id1169739001004\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}f^{\\prime}(x) &amp; =\\frac{d}{dx}(5x^3)\\cdot \\sin x+\\frac{d}{dx}(\\sin x)\\cdot 5x^3 \\\\ &amp; =15x^2\\cdot \\sin x+ \\cos x\\cdot 5x^3\\end{array}[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1169739036358\">After simplifying, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739269866\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=15x^2 \\sin x+5x^3 \\cos x[\/latex].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738980720\" class=\"bc-section section\">\r\n<div id=\"fs-id1169738960853\" class=\"equation unnumbered\" style=\"text-align: center;\">\r\n<div id=\"fs-id1169739300487\" class=\"bc-section section\">\r\n<div id=\"fs-id1169738980720\" class=\"bc-section section\">\r\n<div id=\"fs-id1169739325698\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\"><strong>The Derivatives of\u00a0 [latex]\\tan x, \\, \\cot x, \\, \\sec x[\/latex],\u00a0 and\u00a0 [latex]\\csc x[\/latex]<\/strong><\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1169739299818\">The derivatives of the remaining trigonometric functions are as follows:<\/p>\r\n\r\n<div id=\"fs-id1169739299822\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\tan x)=\\sec^2 x[\/latex]<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<div id=\"fs-id1169739301143\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<div id=\"fs-id1169739301181\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<div id=\"fs-id1169736658480\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739273070\">Find the derivative of [latex]f(x)= \\cot x[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739301461\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739301461\"]\r\n<p id=\"fs-id1169739301461\">[latex]f^{\\prime}(x)=\u2212\\csc^2 x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>Find the Derivatives of Exponential and Logarithmic Functions with Base e<\/h2>\r\n<div id=\"fs-id1169739098813\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Derivative of the Natural Exponential Function<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1169738220224\">Let [latex]E(x)=e^x[\/latex] be the natural exponential function. Then<\/p>\r\n\r\n<div id=\"fs-id1169737928243\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]E^{\\prime}(x)=e^x[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169737142141\">In general,<\/p>\r\n\r\n<div id=\"fs-id1169738124964\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(e^{g(x)})=e^{g(x)} g^{\\prime}(x)[\/latex]<\/div>\r\n<\/div>\r\nIf it helps, think of the formula as the chain rule being applied to natural exponential functions. The derivative of [latex]e[\/latex] raised to the power of a function will simply be [latex]e[\/latex] raised to the power of the function multiplied by the derivative of that function.\r\n<div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Differentiating An Exponential Function<\/h3>\r\n<p id=\"fs-id1169738187154\">Find the derivative of [latex]f(x)=e^{\\tan (2x)}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169737140844\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737140844\"]\r\n<p id=\"fs-id1169737140844\">Using the derivative formula and the chain rule,<\/p>\r\n[latex]\\begin{array}{ll} f^{\\prime}(x) &amp; =e^{\\tan (2x)}\\frac{d}{dx}(\\tan (2x)) \\\\ &amp; = e^{\\tan (2x)} \\sec^2 (2x) \\cdot 2. \\end{array}[\/latex]\r\n\r\nRecall that for exponential functions, we apply the chain rule to the exponent.\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<p id=\"fs-id1169739273070\">Find the derivative of [latex]f(x)=e^{5x^2}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739301845\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739301845\"]\r\n<p id=\"fs-id1169739301461\">[latex]f^{\\prime}(x)=10xe^{5x^2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nNow that we have the derivative of the natural exponential function, we can use implicit differentiation to find the derivative of its inverse, the natural logarithmic function.\r\n<div id=\"fs-id1169738980720\" class=\"bc-section section\">\r\n<div id=\"fs-id1169738960853\" class=\"equation unnumbered\" style=\"text-align: center;\">\r\n<div id=\"fs-id1169739300487\" class=\"bc-section section\">\r\n<div id=\"fs-id1169738980720\" class=\"bc-section section\">\r\n<div id=\"fs-id1169739325698\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">The Derivative of the Natural Logarithmic Function<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1169737911417\">If [latex]x&gt;0[\/latex] and [latex]y=\\ln x[\/latex], then<\/p>\r\n\r\n<div id=\"fs-id1169738223534\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}=\\dfrac{1}{x}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169737765282\">More generally, let [latex]g(x)[\/latex] be a differentiable function. For all values of [latex]x[\/latex] for which [latex]g^{\\prime}(x)&gt;0[\/latex], the derivative of [latex]h(x)=\\ln(g(x))[\/latex] is given by<\/p>\r\n\r\n<div id=\"fs-id1169737919348\" class=\"equation\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=\\dfrac{1}{g(x)} g^{\\prime}(x)[\/latex]<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Differentiating A Natural Logarithmic Function<\/h3>\r\n<p id=\"fs-id1169738228377\">Find the derivative of [latex]f(x)=\\ln(x^3+3x-4)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169738221346\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738221346\"]\r\n\r\nUse the derivative of a natural logarithm directly.\r\n<div id=\"fs-id1169738220266\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll} f^{\\prime}(x) &amp; =\\frac{1}{x^3+3x-4} \\cdot (3x^2+3) &amp; &amp; &amp; \\text{Use} \\, g(x)=x^3+3x-4 \\, \\text{in} \\, h^{\\prime}(x)=\\frac{1}{g(x)} g^{\\prime}(x). \\\\ &amp; =\\frac{3x^2+3}{x^3+3x-4} &amp; &amp; &amp; \\text{Rewrite.} \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<div class=\"equation unnumbered\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739273070\">Find the derivative of [latex]g(x)=\\ln(3x+7)[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739301347\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739301347\"]\r\n<p id=\"fs-id1169739301461\">[latex]g^{\\prime}(x)=\\dfrac{3}{3x+7}[\/latex]<\/p>\r\n[\/hidden-answer]<span style=\"font-size: 1.2em; font-weight: 600; text-align: center; text-transform: uppercase; background-color: #eeeeee;\">\u00a0<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>\u00a0Apply Substitution Integration Shortcut Formulas<\/h2>\r\n<\/div>\r\n<\/div>\r\n<span style=\"font-size: 1rem; text-align: initial;\">When integrating certain functions using substitution, certain patterns can be noticed in the answers. Here, we will review some shortcut integration formulas that are a result of substitution.<\/span>\r\n<div id=\"fs-id1169739098813\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">SUbstitution Integration Shortcut Formulas<\/h3>\r\n\r\n<hr \/>\r\n\r\nLet [latex]a[\/latex] be a constant, then\r\n<ul>\r\n \t<li id=\"fs-id1169738220224\">\r\n<div id=\"fs-id1165043250161\" class=\"equation unnumbered\" style=\"text-align: left;\">[latex]\\displaystyle\\int e^{ax} dx=\\dfrac{e^{ax}}{a}+C[\/latex]<\/div><\/li>\r\n \t<li>\r\n<div id=\"fs-id1165043250161\" class=\"equation unnumbered\" style=\"text-align: left;\">[latex]\\displaystyle\\int \\sin ax dx=-\\dfrac{\\cos ax }{a}+C[\/latex]<\/div><\/li>\r\n \t<li style=\"text-align: left;\">\r\n<div id=\"fs-id1165043250161\" class=\"equation unnumbered\">[latex]\\displaystyle\\int \\cos ax dx=\\dfrac{\\sin ax }{a}+C[\/latex]<\/div>\r\n&nbsp;<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using a Substitution Integration Shortcut Formula<\/h3>\r\n<p id=\"fs-id1169738187154\">Find\u00a0[latex]\\displaystyle\\int e^{10x} dx[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169737140854\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737140854\"]\r\n\r\nUsing the substitution integration shortcut formula\u00a0[latex]\\displaystyle\\int e^{ax} dx=\\dfrac{e^{ax}}{a}+C[\/latex], we have that\r\n\r\n[latex]\\displaystyle\\int e^{10x} dx=\\dfrac{e^{10x}}{10}+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<p id=\"fs-id1169739273070\">Find\u00a0[latex]\\displaystyle\\int \\cos 2x dx[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739301855\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739301855\"]\r\n\r\nUsing the substitution integration shortcut formula [latex]\\displaystyle\\int \\cos ax dx=\\dfrac{\\sin ax }{a}+C[\/latex], we have that\r\n\r\n[latex]\\displaystyle\\int \\cos 2x dx=\\dfrac{\\sin 2x}{2}+C[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Apply the power rule<\/li>\n<li>Find the derivatives of the sine and cosine function.<\/li>\n<li>Find the derivatives of the standard trigonometric functions.<\/li>\n<li>Find the derivative of exponential functions<\/li>\n<li>Find the derivative of logarithmic functions<\/li>\n<li>Apply substitution integration shortcut formulas<\/li>\n<\/ul>\n<\/div>\n<p>In the Integration by Parts section, we will learn how to evaluate integrals where one part of the integral is easily differentiable while the other part is easily integrable. Here we will review some derivative-taking techniques along with substitution integration shortcuts.<\/p>\n<h2>Apply the Power Rule<\/h2>\n<p id=\"fs-id1169739006200\">We know 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 id=\"fs-id1169736619689\">As we shall see, there is a procedure for finding the derivative of the general form [latex]f(x)=x^n[\/latex]. The following theorem states that this\u00a0<strong>power rule<\/strong> holds for all non-variable powers of [latex]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 number. 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<\/div>\n<div>\n<div class=\"textbox exercises\">\n<h3>Example: Applying Basic Derivative Rules<\/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 class=\"textbox 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{Take the derivative term by term.} \\\\ & =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 class=\"textbox exercises\">\n<h3>Example: Applying Basic Derivative Rules<\/h3>\n<p id=\"fs-id1169739300397\">Find the derivative of [latex]f(x)=\\sqrt{x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739064780\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739064780\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739064774\">For any function in radical form, rewrite it in exponential form.<\/p>\n<p>[latex]f(x)=\\sqrt{x}[\/latex] can be rewritten as [latex]f(x)=x^\\frac{1}{2}[\/latex].<\/p>\n<p>Now, take the derivative using the power rule: [latex]f^{\\prime}(x)=\\dfrac{1}{2}x^\\frac{-1}{2}[\/latex].<\/p>\n<div id=\"fs-id1169739273990\" class=\"equation unnumbered\" style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox 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=\"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^{-4}-12x[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739299469\">Find the derivative of [latex]f(x)=\\sqrt{x^7}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736658740\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736658740\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736658726\">Rewrite:[latex]f(x)=x^\\frac{7}{2}[\/latex]<\/p>\n<p>Differentiate: [latex]f^{\\prime}(x)=\\dfrac{7}{2}x^\\frac{5}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<h2>Find the Derivatives of the Standard Trigonometric Functions.<\/h2>\n<div id=\"fs-id1169739098813\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">The Derivatives of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex]<\/h3>\n<hr \/>\n<p id=\"fs-id1169738998734\">The derivative of the sine function is the cosine and the derivative of the cosine function is the negative sine.<\/p>\n<div id=\"fs-id1169738884040\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<div class=\"equation\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex]<\/div>\n<\/div>\n<div>\n<div class=\"textbox exercises\">\n<h3>Example: Differentiating a Function Containing [latex]sinx[\/latex]<\/h3>\n<p id=\"fs-id1169739269454\">Find the derivative of [latex]f(x)=5x^3 \\sin x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739028319\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739028319\" class=\"hidden-answer\" style=\"display: none\">Using the product rule, we have<\/p>\n<div id=\"fs-id1169739001004\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}f^{\\prime}(x) & =\\frac{d}{dx}(5x^3)\\cdot \\sin x+\\frac{d}{dx}(\\sin x)\\cdot 5x^3 \\\\ & =15x^2\\cdot \\sin x+ \\cos x\\cdot 5x^3\\end{array}[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1169739036358\">After simplifying, we obtain<\/p>\n<div id=\"fs-id1169739269866\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=15x^2 \\sin x+5x^3 \\cos x[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738980720\" class=\"bc-section section\">\n<div id=\"fs-id1169738960853\" class=\"equation unnumbered\" style=\"text-align: center;\">\n<div id=\"fs-id1169739300487\" class=\"bc-section section\">\n<div id=\"fs-id1169738980720\" class=\"bc-section section\">\n<div id=\"fs-id1169739325698\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\"><strong>The Derivatives of\u00a0 [latex]\\tan x, \\, \\cot x, \\, \\sec x[\/latex],\u00a0 and\u00a0 [latex]\\csc x[\/latex]<\/strong><\/h3>\n<hr \/>\n<p id=\"fs-id1169739299818\">The derivatives of the remaining trigonometric functions are as follows:<\/p>\n<div id=\"fs-id1169739299822\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\tan x)=\\sec^2 x[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<div id=\"fs-id1169739301143\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<div id=\"fs-id1169739301181\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<div id=\"fs-id1169736658480\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/div>\n<\/div>\n<div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739273070\">Find the derivative of [latex]f(x)= \\cot x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739301461\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739301461\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739301461\">[latex]f^{\\prime}(x)=\u2212\\csc^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Find the Derivatives of Exponential and Logarithmic Functions with Base e<\/h2>\n<div id=\"fs-id1169739098813\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Derivative of the Natural Exponential Function<\/h3>\n<hr \/>\n<p id=\"fs-id1169738220224\">Let [latex]E(x)=e^x[\/latex] be the natural exponential function. Then<\/p>\n<div id=\"fs-id1169737928243\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]E^{\\prime}(x)=e^x[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169737142141\">In general,<\/p>\n<div id=\"fs-id1169738124964\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(e^{g(x)})=e^{g(x)} g^{\\prime}(x)[\/latex]<\/div>\n<\/div>\n<p>If it helps, think of the formula as the chain rule being applied to natural exponential functions. The derivative of [latex]e[\/latex] raised to the power of a function will simply be [latex]e[\/latex] raised to the power of the function multiplied by the derivative of that function.<\/p>\n<div>\n<div class=\"textbox exercises\">\n<h3>Example: Differentiating An Exponential Function<\/h3>\n<p id=\"fs-id1169738187154\">Find the derivative of [latex]f(x)=e^{\\tan (2x)}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169737140844\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169737140844\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737140844\">Using the derivative formula and the chain rule,<\/p>\n<p>[latex]\\begin{array}{ll} f^{\\prime}(x) & =e^{\\tan (2x)}\\frac{d}{dx}(\\tan (2x)) \\\\ & = e^{\\tan (2x)} \\sec^2 (2x) \\cdot 2. \\end{array}[\/latex]<\/p>\n<p>Recall that for exponential functions, we apply the chain rule to the exponent.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739273070\">Find the derivative of [latex]f(x)=e^{5x^2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739301845\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739301845\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739301461\">[latex]f^{\\prime}(x)=10xe^{5x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Now that we have the derivative of the natural exponential function, we can use implicit differentiation to find the derivative of its inverse, the natural logarithmic function.<\/p>\n<div id=\"fs-id1169738980720\" class=\"bc-section section\">\n<div id=\"fs-id1169738960853\" class=\"equation unnumbered\" style=\"text-align: center;\">\n<div id=\"fs-id1169739300487\" class=\"bc-section section\">\n<div id=\"fs-id1169738980720\" class=\"bc-section section\">\n<div id=\"fs-id1169739325698\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">The Derivative of the Natural Logarithmic Function<\/h3>\n<hr \/>\n<p id=\"fs-id1169737911417\">If [latex]x>0[\/latex] and [latex]y=\\ln x[\/latex], then<\/p>\n<div id=\"fs-id1169738223534\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}=\\dfrac{1}{x}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169737765282\">More generally, let [latex]g(x)[\/latex] be a differentiable function. For all values of [latex]x[\/latex] for which [latex]g^{\\prime}(x)>0[\/latex], the derivative of [latex]h(x)=\\ln(g(x))[\/latex] is given by<\/p>\n<div id=\"fs-id1169737919348\" class=\"equation\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=\\dfrac{1}{g(x)} g^{\\prime}(x)[\/latex]<\/div>\n<\/div>\n<div>\n<div>\n<div class=\"textbox exercises\">\n<h3>Example: Differentiating A Natural Logarithmic Function<\/h3>\n<p id=\"fs-id1169738228377\">Find the derivative of [latex]f(x)=\\ln(x^3+3x-4)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738221346\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738221346\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the derivative of a natural logarithm directly.<\/p>\n<div id=\"fs-id1169738220266\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll} f^{\\prime}(x) & =\\frac{1}{x^3+3x-4} \\cdot (3x^2+3) & & & \\text{Use} \\, g(x)=x^3+3x-4 \\, \\text{in} \\, h^{\\prime}(x)=\\frac{1}{g(x)} g^{\\prime}(x). \\\\ & =\\frac{3x^2+3}{x^3+3x-4} & & & \\text{Rewrite.} \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div class=\"equation unnumbered\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739273070\">Find the derivative of [latex]g(x)=\\ln(3x+7)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739301347\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739301347\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739301461\">[latex]g^{\\prime}(x)=\\dfrac{3}{3x+7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 1.2em; font-weight: 600; text-align: center; text-transform: uppercase; background-color: #eeeeee;\">\u00a0<\/span><\/p>\n<\/div>\n<\/div>\n<h2>\u00a0Apply Substitution Integration Shortcut Formulas<\/h2>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 1rem; text-align: initial;\">When integrating certain functions using substitution, certain patterns can be noticed in the answers. Here, we will review some shortcut integration formulas that are a result of substitution.<\/span><\/p>\n<div id=\"fs-id1169739098813\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">SUbstitution Integration Shortcut Formulas<\/h3>\n<hr \/>\n<p>Let [latex]a[\/latex] be a constant, then<\/p>\n<ul>\n<li id=\"fs-id1169738220224\">\n<div id=\"fs-id1165043250161\" class=\"equation unnumbered\" style=\"text-align: left;\">[latex]\\displaystyle\\int e^{ax} dx=\\dfrac{e^{ax}}{a}+C[\/latex]<\/div>\n<\/li>\n<li>\n<div id=\"fs-id1165043250161\" class=\"equation unnumbered\" style=\"text-align: left;\">[latex]\\displaystyle\\int \\sin ax dx=-\\dfrac{\\cos ax }{a}+C[\/latex]<\/div>\n<\/li>\n<li style=\"text-align: left;\">\n<div id=\"fs-id1165043250161\" class=\"equation unnumbered\">[latex]\\displaystyle\\int \\cos ax dx=\\dfrac{\\sin ax }{a}+C[\/latex]<\/div>\n<p>&nbsp;<\/li>\n<\/ul>\n<\/div>\n<div>\n<div class=\"textbox exercises\">\n<h3>Example: Using a Substitution Integration Shortcut Formula<\/h3>\n<p id=\"fs-id1169738187154\">Find\u00a0[latex]\\displaystyle\\int e^{10x} dx[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169737140854\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169737140854\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using the substitution integration shortcut formula\u00a0[latex]\\displaystyle\\int e^{ax} dx=\\dfrac{e^{ax}}{a}+C[\/latex], we have that<\/p>\n<p>[latex]\\displaystyle\\int e^{10x} dx=\\dfrac{e^{10x}}{10}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739273070\">Find\u00a0[latex]\\displaystyle\\int \\cos 2x dx[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739301855\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739301855\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using the substitution integration shortcut formula [latex]\\displaystyle\\int \\cos ax dx=\\dfrac{\\sin ax }{a}+C[\/latex], we have that<\/p>\n<p>[latex]\\displaystyle\\int \\cos 2x dx=\\dfrac{\\sin 2x}{2}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\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-1983\">\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 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