{"id":344,"date":"2021-02-04T01:16:45","date_gmt":"2021-02-04T01:16:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=344"},"modified":"2022-03-16T05:33:22","modified_gmt":"2022-03-16T05:33:22","slug":"finding-the-derivatives-of-trig-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/finding-the-derivatives-of-trig-functions\/","title":{"raw":"Finding the Derivatives of Trig Functions","rendered":"Finding the Derivatives of Trig Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\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>Calculate the higher-order derivatives of the sine and cosine.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169739300487\" class=\"bc-section section\">\r\n<h2>Derivatives of the Sine and Cosine Functions<\/h2>\r\n<p id=\"fs-id1169739274430\">We begin our exploration of the derivative for the sine function by using the formula to make a reasonable guess at its derivative. Recall that for a function [latex]f(x),[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169739055116\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\dfrac{f(x+h)-f(x)}{h}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738837360\">Consequently, for values of [latex]h[\/latex] very close to 0, [latex]f^{\\prime}(x)\\approx \\frac{f(x+h)-f(x)}{h}[\/latex]. We see that by using [latex]h=0.01[\/latex],<\/p>\r\n\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&nbsp;\r\n<p id=\"fs-id1169739223032\">By setting [latex]D(x)=\\frac{\\sin(x+0.01)-\\sin x}{0.01}[\/latex] and using a graphing utility, we can get a graph of an approximation to the derivative of [latex] \\sin x[\/latex] (Figure 1).<\/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 graph of the function [latex]D(x)[\/latex] looks a lot like a cosine curve.[\/caption]\r\n<p id=\"fs-id1169739302416\">Upon inspection, the graph of [latex]D(x)[\/latex] appears to be very close to the graph of the cosine function. Indeed, we will show that<\/p>\r\n\r\n<div id=\"fs-id1169739009210\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/div>\r\n&nbsp;\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\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&nbsp;\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&nbsp;\r\n\r\n<\/div>\r\n<div id=\"fs-id1169738980720\" class=\"bc-section section\">\r\n<h3>Proof<\/h3>\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]. Before beginning, recall two important trigonometric limits we learned in Module 2: Limits.<\/p>\r\n\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&nbsp;\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\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\nWe also recall the following trigonometric identity for the sine of the sum of two angles:\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&nbsp;\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\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[latex]_\\blacksquare[\/latex]\r\n<p id=\"fs-id1169739186572\">Figure 3 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<div class=\"textbook exercises\">\r\n<h3>Example: Differentiating a Function Containing [latex]\\sin x[\/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>\r\n<div id=\"fs-id1169739304325\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169736660715\">Find the derivative of [latex]f(x)= \\sin x \\cos x.[\/latex]<\/p>\r\n[reveal-answer q=\"300277\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"300277\"]\r\n<p id=\"fs-id1169738821957\">Don\u2019t forget to use the product rule.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169739042083\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739042083\"]\r\n<p id=\"fs-id1169739042083\">[latex]f^{\\prime}(x)=\\cos^2 x-\\sin^2 x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try It.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=183&amp;end=247&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesOfTrigonometricFunctions183to247_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.5 Derivatives of Trigonometric Functions (edited)\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]205604[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169738889578\" class=\"textbook exercises\">\r\n<h3>Example: Finding the Derivative of a Function Containing [latex]\\cos x[\/latex]<\/h3>\r\n<p id=\"fs-id1169739229519\">Find the derivative of [latex]g(x)=\\dfrac{\\cos x}{4x^2}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169738969705\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738969705\"]\r\n<p id=\"fs-id1169738969705\">By applying the quotient rule, we have<\/p>\r\n\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&nbsp;\r\n<p id=\"fs-id1169738949562\">Simplifying, we obtain<\/p>\r\n\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[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736615168\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169738971415\">Find the derivative of [latex]f(x)=\\dfrac{x}{\\cos x}[\/latex].<\/p>\r\n[reveal-answer q=\"488399\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"488399\"]\r\n<p id=\"fs-id1169736587923\">Use the quotient rule.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169739286412\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739286412\"]\r\n<p id=\"fs-id1169739286412\">[latex]\\frac{\\cos x+x \\sin x}{\\cos^2 x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try It.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=699&amp;end=766&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesOfTrigonometricFunctions699to766_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.5 Derivatives of Trigonometric Functions (edited)\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1169739220928\" class=\"textbook exercises\">\r\n<h3>Example: An Application to Velocity<\/h3>\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[reveal-answer q=\"fs-id1169739105201\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739105201\"]\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\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&nbsp;\r\n<p id=\"fs-id1169738962954\">so we must solve<\/p>\r\n\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&nbsp;\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]. 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[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739327874\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169738824893\">A particle moves along a coordinate axis. Its position at time [latex]t[\/latex] is given by [latex]s(t)=\\sqrt{3}t+2 \\cos t[\/latex] for [latex]0\\le t\\le 2\\pi[\/latex]. At what times is the particle at rest?<\/p>\r\n[reveal-answer q=\"fs-id1169739274425\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739274425\"]\r\n<p id=\"fs-id1169739274425\">[latex]t=\\frac{\\pi}{3}, \\, t=\\frac{2\\pi}{3}[\/latex]<\/p>\r\n\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169739189964\">Use the previous example as a guide.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Derivatives of Other Trigonometric Functions<\/h2>\r\n<p id=\"fs-id1169739301871\">Since the remaining four trigonometric functions may be expressed as quotients involving sine, cosine, or both, we can use the quotient rule to find formulas for their derivatives.<\/p>\r\n\r\n<div id=\"fs-id1169736589199\" class=\"textbook exercises\">\r\n<h3>Example: The Derivative of the Tangent Function<\/h3>\r\n<p id=\"fs-id1169739303221\">Find the derivative of [latex]f(x)= \\tan x[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739242784\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739242784\"]\r\n<p id=\"fs-id1169739242784\">Start by expressing [latex]\\tan x[\/latex] as the quotient of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex]:<\/p>\r\n\r\n<div id=\"fs-id1169736655821\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(x)= \\tan x=\\dfrac{\\sin x}{\\cos x}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169736615154\">Now apply the quotient rule to obtain<\/p>\r\n\r\n<div id=\"fs-id1169736615157\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\dfrac{\\cos x \\cos x-(\u2212\\sin x)\\sin x}{(\\cos x)^2}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739111151\">Simplifying, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739111154\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\dfrac{\\cos^2 x+\\sin^2 x}{\\cos^2 x}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169736657080\">Recognizing that [latex]\\cos^2 x+\\sin^2 x=1[\/latex], by the Pythagorean Identity, we now have<\/p>\r\n\r\n<div id=\"fs-id1169739190112\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\dfrac{1}{\\cos^2 x}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739225322\">Finally, use the identity [latex]\\sec x=\\dfrac{1}{\\cos x}[\/latex] to obtain<\/p>\r\n\r\n<div id=\"fs-id1169736656627\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\sec^2 x[\/latex]<\/div>\r\n<div class=\"equation unnumbered\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbook 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=\"487336\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"487336\"]\r\n<p id=\"fs-id1169739300242\">Rewrite [latex]\\cot x[\/latex] as [latex]\\frac{\\cos x}{\\sin x}[\/latex] and use the quotient rule.<\/p>\r\n[\/hidden-answer]\r\n\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<p id=\"fs-id1169739325693\">The derivatives of the remaining trigonometric functions may be obtained by using similar techniques. We provide these formulas in the following theorem.<\/p>\r\n\r\n<div id=\"fs-id1169739325698\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\"><strong>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><\/div>\r\n<\/div>\r\nAs you navigate problems involving derivatives of trigonometric functions, don't forget our handy table of trigonometric function values of common angles:\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Trigonometric function values of common angles<\/h3>\r\n<table id=\"Table_05_03_01\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Angle<\/strong><\/td>\r\n<td><strong> [latex]0[\/latex] <\/strong><\/td>\r\n<td><strong> [latex]\\frac{\\pi }{6},\\text{ or }{30}^{\\circ}[\/latex] <\/strong><\/td>\r\n<td><strong> [latex]\\frac{\\pi }{4},\\text{ or } {45}^{\\circ }[\/latex] <\/strong><\/td>\r\n<td><strong> [latex]\\frac{\\pi }{3},\\text{ or }{60}^{\\circ }[\/latex] <\/strong><\/td>\r\n<td><strong> [latex]\\frac{\\pi }{2},\\text{ or }{90}^{\\circ }[\/latex] <\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Cosine<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Sine<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Tangent<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td>Undefined<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Secant<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>Undefined<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Cosecant<\/strong><\/td>\r\n<td>Undefined<\/td>\r\n<td>2<\/td>\r\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Cotangent<\/strong><\/td>\r\n<td>Undefined<\/td>\r\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2><\/h2>\r\n<\/div>\r\n<div id=\"fs-id1169739111296\" class=\"textbook exercises\">\r\n<h3>Example: Finding the Equation of a Tangent Line<\/h3>\r\n<p id=\"fs-id1169739111306\">Find the equation of a line tangent to the graph of [latex]f(x)= \\cot x[\/latex] at [latex]x=\\dfrac{\\pi}{4}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169736658874\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736658874\"]\r\n<p id=\"fs-id1169736658874\">To find the equation of the tangent line, we need a point and a slope at that point. To find the point, compute<\/p>\r\n\r\n<div id=\"fs-id1169736658878\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(\\frac{\\pi}{4}\\right)= \\cot \\frac{\\pi}{4}=1[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169736656804\">Thus the tangent line passes through the point [latex]\\left(\\frac{\\pi}{4},1\\right)[\/latex]. Next, find the slope by finding the derivative of [latex]f(x)= \\cot x[\/latex] and evaluating it at [latex]\\frac{\\pi}{4}[\/latex]:<\/p>\r\n\r\n<div id=\"fs-id1169736589231\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\u2212\\csc^2 x[\/latex]\u00a0 and\u00a0 [latex]f^{\\prime}\\left(\\frac{\\pi}{4}\\right)=\u2212\\csc^2 \\left(\\frac{\\pi}{4}\\right)=-2[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739111215\">Using the point-slope equation of the line, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739111218\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y-1=-2\\left(x-\\frac{\\pi}{4}\\right)[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739336035\">or equivalently,<\/p>\r\n\r\n<div id=\"fs-id1169739336038\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=-2x+1+\\frac{\\pi}{2}[\/latex].<\/div>\r\n&nbsp;\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736655845\" class=\"textbook exercises\">\r\n<h3>Example: Finding the Derivative of Trigonometric Functions<\/h3>\r\n<p id=\"fs-id1169736655855\">Find the derivative of [latex]f(x)= \\csc x+x \\tan x.[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736655897\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736655897\"]\r\n<p id=\"fs-id1169736655897\">To find this derivative, we must use both the sum rule and the product rule. Using the sum rule, we find<\/p>\r\n\r\n<div id=\"fs-id1169736610100\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\frac{d}{dx}(\\csc x)+\\frac{d}{dx}(x \\tan x)[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169736610172\">In the first term, [latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex], and by applying the product rule to the second term we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739182390\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(x \\tan x)=(1)(\\tan x)+(\\sec^2 x)(x)[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739265934\">Therefore, we have<\/p>\r\n\r\n<div id=\"fs-id1169739265938\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\u2212\\csc x \\cot x+ \\tan x+x \\sec^2 x[\/latex].<\/div>\r\n&nbsp;\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\">[\/hidden-answer]<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Finding the Derivative of Trigonometric Functions.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=1247&amp;end=1311&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266835\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266835\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesofTrigonometricFunctions1247to1311_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.5 Derivatives of Trigonometric Functions (edited)\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1169739188143\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739188150\">Find the derivative of [latex]f(x)=2 \\tan x-3 \\cot x[\/latex].<\/p>\r\n[reveal-answer q=\"661109\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"661109\"]\r\n<p id=\"fs-id1169736662619\">Use the rule for differentiating a constant multiple and the rule for differentiating a difference of two functions.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169739188198\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739188198\"]\r\n<p id=\"fs-id1169739188198\">[latex]f^{\\prime}(x)=2 \\sec^2 x+3 \\csc^2 x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736662626\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the slope of the line tangent to the graph of [latex]f(x)= \\tan x[\/latex] at [latex]x=\\frac{\\pi}{6}[\/latex].\r\n\r\n[reveal-answer q=\"6078833\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"6078833\"]\r\n<p id=\"fs-id1169739303758\">Evaluate the derivative at [latex]x=\\frac{\\pi}{6}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169736662675\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736662675\"]\r\n<p id=\"fs-id1169736662675\">[latex]\\frac{4}{3}[\/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[ohm_question]33737[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\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>Calculate the higher-order derivatives of the sine and cosine.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169739300487\" class=\"bc-section section\">\n<h2>Derivatives of the Sine and Cosine Functions<\/h2>\n<p id=\"fs-id1169739274430\">We begin our exploration of the derivative for the sine function by using the formula to make a reasonable guess at its derivative. Recall that for a function [latex]f(x),[\/latex]<\/p>\n<div id=\"fs-id1169739055116\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\dfrac{f(x+h)-f(x)}{h}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738837360\">Consequently, for values of [latex]h[\/latex] very close to 0, [latex]f^{\\prime}(x)\\approx \\frac{f(x+h)-f(x)}{h}[\/latex]. We see that by using [latex]h=0.01[\/latex],<\/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>&nbsp;<\/p>\n<p id=\"fs-id1169739223032\">By setting [latex]D(x)=\\frac{\\sin(x+0.01)-\\sin x}{0.01}[\/latex] and using a graphing utility, we can get a graph of an approximation to the derivative of [latex]\\sin x[\/latex] (Figure 1).<\/p>\n<div style=\"width: 441px\" 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\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. The graph of the function [latex]D(x)[\/latex] looks a lot like a cosine curve.<\/p>\n<\/div>\n<p id=\"fs-id1169739302416\">Upon inspection, the graph of [latex]D(x)[\/latex] appears to be very close to the graph of the cosine function. Indeed, we will show that<\/p>\n<div id=\"fs-id1169739009210\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/div>\n<p>&nbsp;<\/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<p>&nbsp;<\/p>\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<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1169738980720\" class=\"bc-section section\">\n<h3>Proof<\/h3>\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]. Before beginning, recall two important trigonometric limits we learned in Module 2: 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>&nbsp;<\/p>\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<div style=\"width: 810px\" 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\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. These graphs show two important limits needed to establish the derivative formulas for the sine and cosine functions.<\/p>\n<\/div>\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>&nbsp;<\/p>\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]<\/p>\n<p id=\"fs-id1169739186572\">Figure 3 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<div style=\"width: 497px\" 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\" \/><\/p>\n<p 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.<\/p>\n<\/div>\n<div class=\"textbook exercises\">\n<h3>Example: Differentiating a Function Containing [latex]\\sin x[\/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>\n<div id=\"fs-id1169739304325\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169736660715\">Find the derivative of [latex]f(x)= \\sin x \\cos x.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q300277\">Hint<\/span><\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739042083\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739042083\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739042083\">[latex]f^{\\prime}(x)=\\cos^2 x-\\sin^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=183&amp;end=247&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesOfTrigonometricFunctions183to247_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.5 Derivatives of Trigonometric Functions (edited)&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm205604\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=205604&theme=oea&iframe_resize_id=ohm205604&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1169738889578\" class=\"textbook exercises\">\n<h3>Example: Finding the Derivative of a Function Containing [latex]\\cos x[\/latex]<\/h3>\n<p id=\"fs-id1169739229519\">Find the derivative of [latex]g(x)=\\dfrac{\\cos x}{4x^2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738969705\">Show Solution<\/span><\/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>&nbsp;<\/p>\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<\/div>\n<div id=\"fs-id1169736615168\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169738971415\">Find the derivative of [latex]f(x)=\\dfrac{x}{\\cos x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q488399\">Hint<\/span><\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739286412\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739286412\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739286412\">[latex]\\frac{\\cos x+x \\sin x}{\\cos^2 x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=699&amp;end=766&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesOfTrigonometricFunctions699to766_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.5 Derivatives of Trigonometric Functions (edited)&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1169739220928\" class=\"textbook exercises\">\n<h3>Example: An Application to Velocity<\/h3>\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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739105201\">Show Solution<\/span><\/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>&nbsp;<\/p>\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>&nbsp;<\/p>\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]. 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<\/div>\n<div id=\"fs-id1169739327874\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169738824893\">A particle moves along a coordinate axis. Its position at time [latex]t[\/latex] is given by [latex]s(t)=\\sqrt{3}t+2 \\cos t[\/latex] for [latex]0\\le t\\le 2\\pi[\/latex]. At what times is the particle at rest?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739274425\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739274425\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739274425\">[latex]t=\\frac{\\pi}{3}, \\, t=\\frac{2\\pi}{3}[\/latex]<\/p>\n<h4>Hint<\/h4>\n<p id=\"fs-id1169739189964\">Use the previous example as a guide.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Derivatives of Other Trigonometric Functions<\/h2>\n<p id=\"fs-id1169739301871\">Since the remaining four trigonometric functions may be expressed as quotients involving sine, cosine, or both, we can use the quotient rule to find formulas for their derivatives.<\/p>\n<div id=\"fs-id1169736589199\" class=\"textbook exercises\">\n<h3>Example: The Derivative of the Tangent Function<\/h3>\n<p id=\"fs-id1169739303221\">Find the derivative of [latex]f(x)= \\tan x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739242784\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739242784\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739242784\">Start by expressing [latex]\\tan x[\/latex] as the quotient of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex]:<\/p>\n<div id=\"fs-id1169736655821\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(x)= \\tan x=\\dfrac{\\sin x}{\\cos x}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169736615154\">Now apply the quotient rule to obtain<\/p>\n<div id=\"fs-id1169736615157\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\dfrac{\\cos x \\cos x-(\u2212\\sin x)\\sin x}{(\\cos x)^2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739111151\">Simplifying, we obtain<\/p>\n<div id=\"fs-id1169739111154\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\dfrac{\\cos^2 x+\\sin^2 x}{\\cos^2 x}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169736657080\">Recognizing that [latex]\\cos^2 x+\\sin^2 x=1[\/latex], by the Pythagorean Identity, we now have<\/p>\n<div id=\"fs-id1169739190112\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\dfrac{1}{\\cos^2 x}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739225322\">Finally, use the identity [latex]\\sec x=\\dfrac{1}{\\cos x}[\/latex] to obtain<\/p>\n<div id=\"fs-id1169736656627\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\sec^2 x[\/latex]<\/div>\n<div class=\"equation unnumbered\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbook 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=\"q487336\">Hint<\/span><\/p>\n<div id=\"q487336\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739300242\">Rewrite [latex]\\cot x[\/latex] as [latex]\\frac{\\cos x}{\\sin x}[\/latex] and use the quotient 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-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<p id=\"fs-id1169739325693\">The derivatives of the remaining trigonometric functions may be obtained by using similar techniques. We provide these formulas in the following theorem.<\/p>\n<div id=\"fs-id1169739325698\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\"><strong>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><\/div>\n<\/div>\n<p>As you navigate problems involving derivatives of trigonometric functions, don&#8217;t forget our handy table of trigonometric function values of common angles:<\/p>\n<div class=\"textbox examples\">\n<h3>Recall: Trigonometric function values of common angles<\/h3>\n<table id=\"Table_05_03_01\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Angle<\/strong><\/td>\n<td><strong> [latex]0[\/latex] <\/strong><\/td>\n<td><strong> [latex]\\frac{\\pi }{6},\\text{ or }{30}^{\\circ}[\/latex] <\/strong><\/td>\n<td><strong> [latex]\\frac{\\pi }{4},\\text{ or } {45}^{\\circ }[\/latex] <\/strong><\/td>\n<td><strong> [latex]\\frac{\\pi }{3},\\text{ or }{60}^{\\circ }[\/latex] <\/strong><\/td>\n<td><strong> [latex]\\frac{\\pi }{2},\\text{ or }{90}^{\\circ }[\/latex] <\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Cosine<\/strong><\/td>\n<td>1<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td><strong>Sine<\/strong><\/td>\n<td>0<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td><strong>Tangent<\/strong><\/td>\n<td>0<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td>1<\/td>\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\n<td>Undefined<\/td>\n<\/tr>\n<tr>\n<td><strong>Secant<\/strong><\/td>\n<td>1<\/td>\n<td>[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\n<td>2<\/td>\n<td>Undefined<\/td>\n<\/tr>\n<tr>\n<td><strong>Cosecant<\/strong><\/td>\n<td>Undefined<\/td>\n<td>2<\/td>\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\n<td>[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td><strong>Cotangent<\/strong><\/td>\n<td>Undefined<\/td>\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\n<td>1<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2><\/h2>\n<\/div>\n<div id=\"fs-id1169739111296\" class=\"textbook exercises\">\n<h3>Example: Finding the Equation of a Tangent Line<\/h3>\n<p id=\"fs-id1169739111306\">Find the equation of a line tangent to the graph of [latex]f(x)= \\cot x[\/latex] at [latex]x=\\dfrac{\\pi}{4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736658874\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736658874\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736658874\">To find the equation of the tangent line, we need a point and a slope at that point. To find the point, compute<\/p>\n<div id=\"fs-id1169736658878\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(\\frac{\\pi}{4}\\right)= \\cot \\frac{\\pi}{4}=1[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169736656804\">Thus the tangent line passes through the point [latex]\\left(\\frac{\\pi}{4},1\\right)[\/latex]. Next, find the slope by finding the derivative of [latex]f(x)= \\cot x[\/latex] and evaluating it at [latex]\\frac{\\pi}{4}[\/latex]:<\/p>\n<div id=\"fs-id1169736589231\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\u2212\\csc^2 x[\/latex]\u00a0 and\u00a0 [latex]f^{\\prime}\\left(\\frac{\\pi}{4}\\right)=\u2212\\csc^2 \\left(\\frac{\\pi}{4}\\right)=-2[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739111215\">Using the point-slope equation of the line, we obtain<\/p>\n<div id=\"fs-id1169739111218\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y-1=-2\\left(x-\\frac{\\pi}{4}\\right)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739336035\">or equivalently,<\/p>\n<div id=\"fs-id1169739336038\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=-2x+1+\\frac{\\pi}{2}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736655845\" class=\"textbook exercises\">\n<h3>Example: Finding the Derivative of Trigonometric Functions<\/h3>\n<p id=\"fs-id1169736655855\">Find the derivative of [latex]f(x)= \\csc x+x \\tan x.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736655897\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736655897\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736655897\">To find this derivative, we must use both the sum rule and the product rule. Using the sum rule, we find<\/p>\n<div id=\"fs-id1169736610100\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\frac{d}{dx}(\\csc x)+\\frac{d}{dx}(x \\tan x)[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169736610172\">In the first term, [latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex], and by applying the product rule to the second term we obtain<\/p>\n<div id=\"fs-id1169739182390\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(x \\tan x)=(1)(\\tan x)+(\\sec^2 x)(x)[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739265934\">Therefore, we have<\/p>\n<div id=\"fs-id1169739265938\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\u2212\\csc x \\cot x+ \\tan x+x \\sec^2 x[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Finding the Derivative of Trigonometric Functions.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=1247&amp;end=1311&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266835\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266835\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesofTrigonometricFunctions1247to1311_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.5 Derivatives of Trigonometric Functions (edited)&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1169739188143\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739188150\">Find the derivative of [latex]f(x)=2 \\tan x-3 \\cot x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q661109\">Hint<\/span><\/p>\n<div id=\"q661109\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736662619\">Use the rule for differentiating a constant multiple and the rule for differentiating a difference of two functions.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739188198\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739188198\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739188198\">[latex]f^{\\prime}(x)=2 \\sec^2 x+3 \\csc^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736662626\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the slope of the line tangent to the graph of [latex]f(x)= \\tan x[\/latex] at [latex]x=\\frac{\\pi}{6}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q6078833\">Hint<\/span><\/p>\n<div id=\"q6078833\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739303758\">Evaluate the derivative at [latex]x=\\frac{\\pi}{6}[\/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-id1169736662675\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736662675\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736662675\">[latex]\\frac{4}{3}[\/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=\"ohm33737\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=33737&theme=oea&iframe_resize_id=ohm33737&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-344\">\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>3.5 Derivatives of Trigonometric Functions (edited). <strong>Authored by<\/strong>: Ryan Melton. <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>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":21,"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\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-344","chapter","type-chapter","status-publish","hentry"],"part":35,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/344","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":27,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/344\/revisions"}],"predecessor-version":[{"id":4811,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/344\/revisions\/4811"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/35"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/344\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=344"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=344"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=344"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=344"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}