{"id":248,"date":"2023-09-20T22:48:33","date_gmt":"2023-09-20T22:48:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/finding-the-derivatives-of-trig-functions\/"},"modified":"2024-08-05T12:45:08","modified_gmt":"2024-08-05T12:45:08","slug":"derivatives-of-trigonometric-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/derivatives-of-trigonometric-functions-learn-it-1\/","title":{"raw":"Derivatives of Trigonometric Functions: Learn It 1","rendered":"Derivatives of Trigonometric Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Calculate the derivatives of sine and cosine functions, including second derivatives and beyond<\/li>\r\n\t<li>Determine the derivatives for basic trig functions like tangent, cotangent, secant, and cosecant<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Derivatives of the Sine and Cosine Functions<\/h2>\r\n<p>Simple harmonic motion, a type of periodic motion where the restoring force is directly proportional to the displacement, is best described using trigonometric functions like sine and cosine. The behavior of these functions, particularly how they change over time, is crucial in understanding motion dynamics. The derivatives of sine and cosine functions help us compute velocity and acceleration at any point in the motion, linking theoretical physics closely with calculus.<\/p>\r\n<p id=\"fs-id1169739274430\">We begin our exploration of the derivative for the sine function by using the limit definition to estimate its derivative.<\/p>\r\n<section class=\"textbox recall\">\r\n<p>For a function [latex]f(x),[\/latex] the derivative [latex]f^{\\prime}(x)[\/latex] is defined as:<\/p>\r\n<center>[latex]f^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\dfrac{f(x+h)-f(x)}{h}[\/latex]<\/center>\r\n<p>This allows us to approximate [latex]f^{\\prime}(x)[\/latex] for small values of [latex]h[\/latex] as:<\/p>\r\n<center>[latex]f^{\\prime}(x)\\approx \\frac{f(x+h)-f(x)}{h}[\/latex].<\/center><\/section>\r\n<p id=\"fs-id1169738837360\">Using [latex]h=0.01[\/latex], we estimate the derivative of the sine function as follows:\u00a0<\/p>\r\n<div id=\"fs-id1169738997799\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\sin x)\\approx \\dfrac{\\sin(x+0.01)-\\sin x}{0.01}[\/latex]<\/div>\r\n<p id=\"fs-id1169739223032\">By defining [latex]D(x)=\\frac{\\sin(x+0.01)-\\sin x}{0.01}[\/latex] and plotting this using a graphing tool, we observe an approximation to the derivative of [latex] \\sin x[\/latex].\u00a0<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"431\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205417\/CNX_Calc_Figure_03_05_001.jpg\" alt=\"The function D(x) = (sin(x + 0.01) \u2212 sin x)\/0.01 is graphed. It looks a lot like a cosine curve.\" width=\"431\" height=\"392\" \/> Figure 1. The resulting graph of [latex]D(x)[\/latex] closely resembles the cosine curve, which supports the derivative relationship.[\/caption]\r\n\r\n<p id=\"fs-id1169739302416\">Upon examination, [latex]D(x)[\/latex] appears to be a close match to the graph of the cosine function. This graphical analysis provides a practical demonstration of the derivative, confirming that the derivative of [latex] \\sin x[\/latex] is indeed [latex] \\cos x[\/latex].<\/p>\r\n<p id=\"fs-id1169738975910\">If we were to follow the same steps to approximate the derivative of the cosine function, we would find that<\/p>\r\n<div id=\"fs-id1169738962070\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex]<\/div>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3 style=\"text-align: left;\">derivatives of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex]<\/h3>\r\n<p id=\"fs-id1169738998734\">The derivative of the sine function [latex] \\sin x[\/latex] is the cosine function [latex] \\cos x[\/latex].<\/p>\r\n<center>[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/center>The derivative of the cosine function [latex] \\cos x[\/latex] is the negative sine function [latex]\u2212\\sin x[\/latex].<center>[latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex]<\/center><\/section>\r\n<section class=\"textbox connectIt\">\r\n<p style=\"text-align: center;\"><strong>Proof<\/strong><\/p>\r\n<hr \/>\r\n<p id=\"fs-id1169738916826\">Because the proofs for [latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex] and [latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex] use similar techniques, we provide only the proof for [latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex].<\/p>\r\n<p>Before beginning, it is important to recall two important trigonometric limits:<\/p>\r\n<div id=\"fs-id1169738960853\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{h\\to 0}{\\lim}\\frac{\\sin h}{h}=1[\/latex]\u00a0 and\u00a0 [latex]\\underset{h\\to 0}{\\lim}\\frac{\\cos h-1}{h}=0[\/latex]<\/div>\r\n<p id=\"fs-id1169739273494\">The graphs of [latex]y=\\frac{(\\sin h)}{h}[\/latex] and [latex]y=\\frac{(\\cos h-1)}{h}[\/latex] are shown in Figure 2.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"800\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205419\/CNX_Calc_Figure_03_05_002.jpg\" alt=\"The function y = (sin h)\/h and y = (cos h \u2013 1)\/h are graphed. They both have discontinuities on the y-axis.\" width=\"800\" height=\"386\" \/> Figure 2. These graphs show two important limits needed to establish the derivative formulas for the sine and cosine functions.[\/caption]\r\n\r\n<p>We also recall the following trigonometric identity for the sine of the sum of two angles:<\/p>\r\n<div id=\"fs-id1169739027491\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex] \\sin(x+h)= \\sin x \\cos h+ \\cos x \\sin h[\/latex]<\/div>\r\n<p id=\"fs-id1169739036876\">Now that we have gathered all the necessary equations and identities, we proceed with the proof.<\/p>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}\\frac{d}{dx} \\sin x &amp; =\\underset{h\\to 0}{\\lim}\\frac{\\sin(x+h)-\\sin x}{h} &amp; &amp; &amp; \\text{Apply the definition of the derivative.} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{\\sin x \\cos h+ \\cos x \\sin h- \\sin x}{h} &amp; &amp; &amp; \\text{Use trig identity for the sine of the sum of two angles.} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\left(\\frac{\\sin x \\cos h-\\sin x}{h}+\\frac{\\cos x \\sin h}{h}\\right) &amp; &amp; &amp; \\text{Regroup.} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\left(\\sin x\\left(\\frac{\\cos h-1}{h}\\right)+ \\cos x\\left(\\frac{\\sin h}{h}\\right)\\right) &amp; &amp; &amp; \\text{Factor out} \\, \\sin x \\, \\text{and} \\, \\cos x. \\\\ &amp; = \\sin x\\cdot{0}+ \\cos x\\cdot{1} &amp; &amp; &amp; \\text{Apply trig limit formulas.} \\\\ &amp; = \\cos x &amp; &amp; &amp; \\text{Simplify.} \\end{array}[\/latex]<\/div>\r\n<p>[latex]_\\blacksquare[\/latex]<\/p>\r\n<\/section>\r\n<p id=\"fs-id1169739186572\">The figure below shows the relationship between the graph of [latex]f(x)= \\sin x[\/latex] and its derivative [latex]f^{\\prime}(x)= \\cos x[\/latex]. Notice that at the points where [latex]f(x)= \\sin x[\/latex] has a horizontal tangent, its derivative [latex]f^{\\prime}(x)= \\cos x[\/latex] takes on the value zero. We also see that where [latex]f(x)= \\sin x[\/latex] is increasing, [latex]f^{\\prime}(x)= \\cos x&gt;0[\/latex] and where [latex]f(x)= \\sin x[\/latex] is decreasing, [latex]f^{\\prime}(x)= \\cos x&lt;0[\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205423\/CNX_Calc_Figure_03_05_003.jpg\" alt=\"The functions f(x) = sin x and f\u2019(x) = cos x are graphed. It is apparent that when f(x) has a maximum or a minimum that f\u2019(x) = 0.\" width=\"487\" height=\"358\" \/> Figure 3. Where [latex]f(x)[\/latex] has a maximum or a minimum, [latex]f^{\\prime}(x)=0[\/latex]. That is, [latex]f^{\\prime}(x)=0[\/latex] where [latex]f(x)[\/latex] has a horizontal tangent. These points are noted with dots on the graphs.[\/caption]\r\n\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739269454\">Find the derivative of [latex]f(x)=5x^3 \\sin x[\/latex].<\/p>\r\n<p>[reveal-answer q=\"300277\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"300277\"]<\/p>\r\n<p id=\"fs-id1169738821957\">Don\u2019t forget to use the product rule.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739028319\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739028319\"]Using the product rule, we have<\/p>\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<p id=\"fs-id1169739036358\">After simplifying, we obtain<\/p>\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<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739229519\">Find the derivative of [latex]g(x)=\\dfrac{\\cos x}{4x^2}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"488399\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"488399\"]<\/p>\r\n<p id=\"fs-id1169736587923\">Use the quotient rule.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169738969705\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169738969705\"]<\/p>\r\n<p id=\"fs-id1169738969705\">By applying the quotient rule, we have<\/p>\r\n<div id=\"fs-id1169738960497\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]g^{\\prime}(x)=\\frac{(\u2212\\sin x)4x^2-8x(\\cos x)}{(4x^2)^2}[\/latex].<\/div>\r\n<p id=\"fs-id1169738949562\">Simplifying, we obtain<\/p>\r\n<div id=\"fs-id1169739179211\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}g^{\\prime}(x) &amp; =\\frac{-4x^2 \\sin x-8x \\cos x}{16x^4} \\\\ &amp; =\\frac{\u2212x \\sin x-2 \\cos x}{4x^3} \\end{array}[\/latex]<\/div>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]205604[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739298225\">A particle moves along a coordinate axis in such a way that its position at time [latex]t[\/latex] is given by [latex]s(t)=2 \\sin t-t[\/latex] for [latex]0\\le t\\le 2\\pi[\/latex]. At what times is the particle at rest?<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739105201\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739105201\"]<\/p>\r\n<p id=\"fs-id1169739105201\">To determine when the particle is at rest, set [latex]s^{\\prime}(t)=v(t)=0[\/latex]. Begin by finding [latex]s^{\\prime}(t)[\/latex]. We obtain<\/p>\r\n<div id=\"fs-id1169739000177\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]s^{\\prime}(t)=2 \\cos t-1[\/latex],<\/div>\r\n<p id=\"fs-id1169738962954\">so we must solve<\/p>\r\n<div id=\"fs-id1169739188455\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]2 \\cos t-1=0[\/latex] for [latex]0\\le t\\le 2\\pi[\/latex].<\/div>\r\n<p id=\"fs-id1169739300388\">The solutions to this equation are [latex]t=\\frac{\\pi}{3}[\/latex] and [latex]t=\\frac{5\\pi}{3}[\/latex].\u00a0<\/p>\r\n<p>Thus the particle is at rest at times [latex]t=\\frac{\\pi}{3}[\/latex] and [latex]t=\\frac{5\\pi}{3}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate the derivatives of sine and cosine functions, including second derivatives and beyond<\/li>\n<li>Determine the derivatives for basic trig functions like tangent, cotangent, secant, and cosecant<\/li>\n<\/ul>\n<\/section>\n<h2>Derivatives of the Sine and Cosine Functions<\/h2>\n<p>Simple harmonic motion, a type of periodic motion where the restoring force is directly proportional to the displacement, is best described using trigonometric functions like sine and cosine. The behavior of these functions, particularly how they change over time, is crucial in understanding motion dynamics. The derivatives of sine and cosine functions help us compute velocity and acceleration at any point in the motion, linking theoretical physics closely with calculus.<\/p>\n<p id=\"fs-id1169739274430\">We begin our exploration of the derivative for the sine function by using the limit definition to estimate its derivative.<\/p>\n<section class=\"textbox recall\">\n<p>For a function [latex]f(x),[\/latex] the derivative [latex]f^{\\prime}(x)[\/latex] is defined as:<\/p>\n<div style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\dfrac{f(x+h)-f(x)}{h}[\/latex]<\/div>\n<p>This allows us to approximate [latex]f^{\\prime}(x)[\/latex] for small values of [latex]h[\/latex] as:<\/p>\n<div style=\"text-align: center;\">[latex]f^{\\prime}(x)\\approx \\frac{f(x+h)-f(x)}{h}[\/latex].<\/div>\n<\/section>\n<p id=\"fs-id1169738837360\">Using [latex]h=0.01[\/latex], we estimate the derivative of the sine function as follows:\u00a0<\/p>\n<div id=\"fs-id1169738997799\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\sin x)\\approx \\dfrac{\\sin(x+0.01)-\\sin x}{0.01}[\/latex]<\/div>\n<p id=\"fs-id1169739223032\">By defining [latex]D(x)=\\frac{\\sin(x+0.01)-\\sin x}{0.01}[\/latex] and plotting this using a graphing tool, we observe an approximation to the derivative of [latex]\\sin x[\/latex].\u00a0<\/p>\n<figure style=\"width: 431px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205417\/CNX_Calc_Figure_03_05_001.jpg\" alt=\"The function D(x) = (sin(x + 0.01) \u2212 sin x)\/0.01 is graphed. It looks a lot like a cosine curve.\" width=\"431\" height=\"392\" \/><figcaption class=\"wp-caption-text\">Figure 1. The resulting graph of [latex]D(x)[\/latex] closely resembles the cosine curve, which supports the derivative relationship.<\/figcaption><\/figure>\n<p id=\"fs-id1169739302416\">Upon examination, [latex]D(x)[\/latex] appears to be a close match to the graph of the cosine function. This graphical analysis provides a practical demonstration of the derivative, confirming that the derivative of [latex]\\sin x[\/latex] is indeed [latex]\\cos x[\/latex].<\/p>\n<p id=\"fs-id1169738975910\">If we were to follow the same steps to approximate the derivative of the cosine function, we would find that<\/p>\n<div id=\"fs-id1169738962070\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex]<\/div>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">derivatives of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex]<\/h3>\n<p id=\"fs-id1169738998734\">The derivative of the sine function [latex]\\sin x[\/latex] is the cosine function [latex]\\cos x[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/div>\n<p>The derivative of the cosine function [latex]\\cos x[\/latex] is the negative sine function [latex]\u2212\\sin x[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex]<\/div>\n<\/section>\n<section class=\"textbox connectIt\">\n<p style=\"text-align: center;\"><strong>Proof<\/strong><\/p>\n<hr \/>\n<p id=\"fs-id1169738916826\">Because the proofs for [latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex] and [latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex] use similar techniques, we provide only the proof for [latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex].<\/p>\n<p>Before beginning, it is important to recall two important trigonometric limits:<\/p>\n<div id=\"fs-id1169738960853\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{h\\to 0}{\\lim}\\frac{\\sin h}{h}=1[\/latex]\u00a0 and\u00a0 [latex]\\underset{h\\to 0}{\\lim}\\frac{\\cos h-1}{h}=0[\/latex]<\/div>\n<p id=\"fs-id1169739273494\">The graphs of [latex]y=\\frac{(\\sin h)}{h}[\/latex] and [latex]y=\\frac{(\\cos h-1)}{h}[\/latex] are shown in Figure 2.<\/p>\n<figure style=\"width: 800px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205419\/CNX_Calc_Figure_03_05_002.jpg\" alt=\"The function y = (sin h)\/h and y = (cos h \u2013 1)\/h are graphed. They both have discontinuities on the y-axis.\" width=\"800\" height=\"386\" \/><figcaption class=\"wp-caption-text\">Figure 2. These graphs show two important limits needed to establish the derivative formulas for the sine and cosine functions.<\/figcaption><\/figure>\n<p>We also recall the following trigonometric identity for the sine of the sum of two angles:<\/p>\n<div id=\"fs-id1169739027491\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sin(x+h)= \\sin x \\cos h+ \\cos x \\sin h[\/latex]<\/div>\n<p id=\"fs-id1169739036876\">Now that we have gathered all the necessary equations and identities, we proceed with the proof.<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}\\frac{d}{dx} \\sin x & =\\underset{h\\to 0}{\\lim}\\frac{\\sin(x+h)-\\sin x}{h} & & & \\text{Apply the definition of the derivative.} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{\\sin x \\cos h+ \\cos x \\sin h- \\sin x}{h} & & & \\text{Use trig identity for the sine of the sum of two angles.} \\\\ & =\\underset{h\\to 0}{\\lim}\\left(\\frac{\\sin x \\cos h-\\sin x}{h}+\\frac{\\cos x \\sin h}{h}\\right) & & & \\text{Regroup.} \\\\ & =\\underset{h\\to 0}{\\lim}\\left(\\sin x\\left(\\frac{\\cos h-1}{h}\\right)+ \\cos x\\left(\\frac{\\sin h}{h}\\right)\\right) & & & \\text{Factor out} \\, \\sin x \\, \\text{and} \\, \\cos x. \\\\ & = \\sin x\\cdot{0}+ \\cos x\\cdot{1} & & & \\text{Apply trig limit formulas.} \\\\ & = \\cos x & & & \\text{Simplify.} \\end{array}[\/latex]<\/div>\n<p>[latex]_\\blacksquare[\/latex]<br \/>\n<\/section>\n<p id=\"fs-id1169739186572\">The figure below shows the relationship between the graph of [latex]f(x)= \\sin x[\/latex] and its derivative [latex]f^{\\prime}(x)= \\cos x[\/latex]. Notice that at the points where [latex]f(x)= \\sin x[\/latex] has a horizontal tangent, its derivative [latex]f^{\\prime}(x)= \\cos x[\/latex] takes on the value zero. We also see that where [latex]f(x)= \\sin x[\/latex] is increasing, [latex]f^{\\prime}(x)= \\cos x>0[\/latex] and where [latex]f(x)= \\sin x[\/latex] is decreasing, [latex]f^{\\prime}(x)= \\cos x<0[\/latex].<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205423\/CNX_Calc_Figure_03_05_003.jpg\" alt=\"The functions f(x) = sin x and f\u2019(x) = cos x are graphed. It is apparent that when f(x) has a maximum or a minimum that f\u2019(x) = 0.\" width=\"487\" height=\"358\" \/><figcaption class=\"wp-caption-text\">Figure 3. Where [latex]f(x)[\/latex] has a maximum or a minimum, [latex]f^{\\prime}(x)=0[\/latex]. That is, [latex]f^{\\prime}(x)=0[\/latex] where [latex]f(x)[\/latex] has a horizontal tangent. These points are noted with dots on the graphs.<\/figcaption><\/figure>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739269454\">Find the derivative of [latex]f(x)=5x^3 \\sin x[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q300277\">Hint<\/button><\/p>\n<div id=\"q300277\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738821957\">Don\u2019t forget to use the product rule.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739028319\">Show Solution<\/button><\/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<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<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739229519\">Find the derivative of [latex]g(x)=\\dfrac{\\cos x}{4x^2}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q488399\">Hint<\/button><\/p>\n<div id=\"q488399\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736587923\">Use the quotient rule.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738969705\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738969705\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738969705\">By applying the quotient rule, we have<\/p>\n<div id=\"fs-id1169738960497\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]g^{\\prime}(x)=\\frac{(\u2212\\sin x)4x^2-8x(\\cos x)}{(4x^2)^2}[\/latex].<\/div>\n<p id=\"fs-id1169738949562\">Simplifying, we obtain<\/p>\n<div id=\"fs-id1169739179211\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}g^{\\prime}(x) & =\\frac{-4x^2 \\sin x-8x \\cos x}{16x^4} \\\\ & =\\frac{\u2212x \\sin x-2 \\cos x}{4x^3} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm205604\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=205604&theme=lumen&iframe_resize_id=ohm205604&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739298225\">A particle moves along a coordinate axis in such a way that its position at time [latex]t[\/latex] is given by [latex]s(t)=2 \\sin t-t[\/latex] for [latex]0\\le t\\le 2\\pi[\/latex]. At what times is the particle at rest?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739105201\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739105201\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739105201\">To determine when the particle is at rest, set [latex]s^{\\prime}(t)=v(t)=0[\/latex]. Begin by finding [latex]s^{\\prime}(t)[\/latex]. We obtain<\/p>\n<div id=\"fs-id1169739000177\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]s^{\\prime}(t)=2 \\cos t-1[\/latex],<\/div>\n<p id=\"fs-id1169738962954\">so we must solve<\/p>\n<div id=\"fs-id1169739188455\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]2 \\cos t-1=0[\/latex] for [latex]0\\le t\\le 2\\pi[\/latex].<\/div>\n<p id=\"fs-id1169739300388\">The solutions to this equation are [latex]t=\\frac{\\pi}{3}[\/latex] and [latex]t=\\frac{5\\pi}{3}[\/latex].\u00a0<\/p>\n<p>Thus the particle is at rest at times [latex]t=\\frac{\\pi}{3}[\/latex] and [latex]t=\\frac{5\\pi}{3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"3.5 Derivatives of Trigonometric Functions (edited)\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":560,"module-header":"learn_it","content_attributions":[{"type":"cc","description":"Calculus Volume 1","author":"Gilbert Strang, Edwin (Jed) Herman","organization":"OpenStax","url":"https:\/\/openstax.org\/details\/books\/calculus-volume-1","project":"","license":"cc-by-nc-sa","license_terms":"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction"},{"type":"original","description":"3.5 Derivatives of Trigonometric Functions (edited)","author":"Ryan Melton","organization":"","url":"","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/248"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/248\/revisions"}],"predecessor-version":[{"id":4512,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/248\/revisions\/4512"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/560"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/248\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=248"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=248"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=248"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=248"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}