{"id":1847,"date":"2018-01-11T20:54:31","date_gmt":"2018-01-11T20:54:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/derivatives-of-trigonometric-functions\/"},"modified":"2018-02-05T15:57:51","modified_gmt":"2018-02-05T15:57:51","slug":"derivatives-of-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/chapter\/derivatives-of-trigonometric-functions\/","title":{"raw":"3.5 Derivatives of Trigonometric Functions","rendered":"3.5 Derivatives of Trigonometric Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/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\nOne of the most important types of motion in physics is simple harmonic motion, which is associated with such systems as an object with mass oscillating on a spring. Simple harmonic motion can be described by using either sine or cosine functions. In this section we expand our knowledge of derivative formulas to include derivatives of these and other trigonometric functions. We begin with the derivatives of the sine and cosine functions and then use them to obtain formulas for the derivatives of the remaining four trigonometric functions. Being able to calculate the derivatives of the sine and cosine functions will enable us to find the velocity and acceleration of simple harmonic motion.\r\n<div id=\"fs-id1169739300487\" class=\"bc-section section\">\r\n<h1>Derivatives of the Sine and Cosine Functions<\/h1>\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\">[latex]{f}^{\\prime }(x)=\\underset{h\\to 0}{\\text{lim}}\\frac{f(x+h)-f(x)}{h}.[\/latex]<\/div>\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\">[latex]\\frac{d}{dx}( \\sin x)\\approx \\frac{ \\sin (x+0.01)- \\sin x}{0.01}[\/latex]<\/div>\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] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_03_05_001\">(Figure)<\/a>).<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_03_05_001\" class=\"wp-caption aligncenter\">[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\" \/> <strong>Figure 1.<\/strong> The graph of the function [latex]D(x)[\/latex] looks a lot like a cosine curve.[\/caption]<\/div>\r\n<div class=\"wp-caption-text\"><\/div>\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\">[latex]\\frac{d}{dx}( \\sin x)= \\cos x.[\/latex]<\/div>\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\">[latex]\\frac{d}{dx}( \\cos x)=\\text{\u2212} \\sin \\mathrm{x.}[\/latex]<\/div>\r\n<div id=\"fs-id1169739098813\" class=\"textbox key-takeaways theorem\">\r\n<h3>The Derivatives of sin [latex]x[\/latex] and cos [latex]x[\/latex]<\/h3>\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\">[latex]\\frac{d}{dx}( \\sin x)= \\cos x[\/latex]<\/div>\r\n<div class=\"equation\">[latex]\\frac{d}{dx}( \\cos x)=\\text{\u2212} \\sin x[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738980720\" class=\"bc-section section\">\r\n<h2>Proof<\/h2>\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)=\\text{\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 <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction-2\/\">Introduction to Limits<\/a>:<\/p>\r\n\r\n<div id=\"fs-id1169738960853\" class=\"equation unnumbered\">[latex]\\underset{h\\to 0}{\\text{lim}}\\frac{ \\sin h}{h}=1\\text{ and }\\underset{h\\to 0}{\\text{lim}}\\frac{\\text{cosh}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 <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_03_05_002\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_03_05_002\" class=\"wp-caption aligncenter\">\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<\/div>\r\n<div class=\"wp-caption-text\"><\/div>\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\">[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\r\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ccccc}\\hfill \\frac{d}{dx} \\sin x&amp; =\\underset{h\\to 0}{\\text{lim}}\\frac{ \\sin (x+h)- \\sin x}{h}\\hfill &amp; &amp; &amp; \\text{Apply the definition of the derivative.}\\hfill \\\\ &amp; =\\underset{h\\to 0}{\\text{lim}}\\frac{ \\sin x \\cos h+ \\cos x \\sin h- \\sin x}{h}\\hfill &amp; &amp; &amp; \\text{Use trig identity for the sine of the sum of two angles.}\\hfill \\\\ &amp; =\\underset{h\\to 0}{\\text{lim}}(\\frac{ \\sin x \\cos h- \\sin x}{h}+\\frac{ \\cos x \\sin h}{h})\\hfill &amp; &amp; &amp; \\text{Regroup.}\\hfill \\\\ &amp; =\\underset{h\\to 0}{\\text{lim}}( \\sin x(\\frac{ \\cos xh-1}{h})+ \\cos x(\\frac{ \\sin h}{h}))\\hfill &amp; &amp; &amp; \\text{Factor out} \\sin x\\text{ and } \\cos x.\\hfill \\\\ &amp; = \\sin x(0)+ \\cos x(1)\\hfill &amp; &amp; &amp; \\text{Apply trig limit formulas.}\\hfill \\\\ &amp; = \\cos x\\hfill &amp; &amp; &amp; \\text{Simplify.}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1169739009931\">\u25a1<\/p>\r\n<p id=\"fs-id1169739186572\"><a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_03_05_003\">(Figure)<\/a> 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<div id=\"CNX_Calc_Figure_03_05_003\" class=\"wp-caption aligncenter\">[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]<\/div>\r\n<div class=\"wp-caption-text\"><\/div>\r\n<div class=\"textbox examples\">\r\n<div id=\"fs-id1169738907624\" class=\"exercise\">\r\n<div id=\"fs-id1169739015112\" class=\"textbox\">\r\n<h3>Differentiating a Function Containing sin [latex]x[\/latex]<\/h3>\r\n<p id=\"fs-id1169739269454\">Find the derivative of [latex]f(x)=5{x}^{3} \\sin x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739028319\" class=\"textbox shaded\">[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\">[latex]\\begin{array}{cc}\\hfill f\\prime (x)&amp; =\\frac{d}{dx}(5{x}^{3})\u00b7 \\sin x+\\frac{d}{dx}( \\sin x)\u00b75{x}^{3}\\hfill \\\\ &amp; =15{x}^{2}\u00b7 \\sin x+ \\cos x\u00b75{x}^{3}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1169739036358\">After simplifying, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739269866\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=15{x}^{2} \\sin x+5{x}^{3} \\cos x.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739304325\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169739327353\" class=\"exercise\">\r\n<div id=\"fs-id1169739009945\" class=\"textbox\">\r\n<p id=\"fs-id1169736660715\">Find the derivative of [latex]f(x)= \\sin x \\cos x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[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\r\n<\/div>\r\n<div id=\"fs-id1169736614177\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169738821957\">Don\u2019t forget to use the product rule.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738889578\" class=\"textbox examples\">\r\n<h3>Finding the Derivative of a Function Containing cos [latex]x[\/latex]<\/h3>\r\n<div id=\"fs-id1169736615213\" class=\"exercise\">\r\n<div id=\"fs-id1169738994002\" class=\"textbox\">\r\n<p id=\"fs-id1169739229519\">Find the derivative of [latex]g(x)=\\frac{ \\cos x}{4{x}^{2}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[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\">[latex]{g}^{\\prime }(x)=\\frac{(\\text{\u2212} \\sin x)4{x}^{2}-8x( \\cos x)}{{(4{x}^{2})}^{2}}.[\/latex]<\/div>\r\n<p id=\"fs-id1169738949562\">Simplifying, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739179211\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill {g}^{\\prime }(x)&amp; =\\frac{-4{x}^{2} \\sin x-8x \\cos x}{16{x}^{4}}\\hfill \\\\ &amp; =\\frac{\\text{\u2212}x \\sin x-2 \\cos x}{4{x}^{3}}.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736615168\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169738971411\" class=\"exercise\">\r\n<div id=\"fs-id1169738971413\" class=\"textbox\">\r\n<p id=\"fs-id1169738971415\">Find the derivative of [latex]f(x)=\\frac{x}{ \\cos x}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[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\r\n<\/div>\r\n<div id=\"fs-id1169736654818\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169736587923\">Use the quotient rule.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739220928\" class=\"textbox examples\">\r\n<h3>An Application to Velocity<\/h3>\r\n<div id=\"fs-id1169739220930\" class=\"exercise\">\r\n<div id=\"fs-id1169739220933\" class=\"textbox\">\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\r\n<\/div>\r\n<div class=\"textbox shaded\">[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\">[latex]{s}^{\\prime }(t)=2 \\cos t-1,[\/latex]<\/div>\r\n<p id=\"fs-id1169738962954\">so we must solve<\/p>\r\n\r\n<div id=\"fs-id1169739188455\" class=\"equation unnumbered\">[latex]2 \\cos t-1=0\\text{ for }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] 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>\r\n<\/div>\r\n<div id=\"fs-id1169739327874\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169739327877\" class=\"exercise\">\r\n<div id=\"fs-id1169738824891\" class=\"textbox\">\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\r\n<\/div>\r\n<div class=\"textbox shaded\">[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<\/div>\r\n<div id=\"fs-id1169739189957\" class=\"commentary\">\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<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736662797\" class=\"bc-section section\">\r\n<h1>Derivatives of Other Trigonometric Functions<\/h1>\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=\"textbox examples\">\r\n<h3>The Derivative of the Tangent Function<\/h3>\r\n<div id=\"fs-id1169736589201\" class=\"exercise\">\r\n<div id=\"fs-id1169736589203\" class=\"textbox\">\r\n<p id=\"fs-id1169739303221\">Find the derivative of [latex]f(x)= \\tan x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[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\">[latex]f(x)= \\tan x=\\frac{ \\sin x}{ \\cos x}.[\/latex]<\/div>\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\">[latex]{f}^{\\prime }(x)=\\frac{ \\cos x \\cos x-(\\text{\u2212} \\sin x) \\sin x}{{( \\cos x)}^{2}}.[\/latex]<\/div>\r\n<p id=\"fs-id1169739111151\">Simplifying, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739111154\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\frac{{ \\cos }^{2}x+{ \\sin }^{2}x}{{ \\cos }^{2}x}.[\/latex]<\/div>\r\n<p id=\"fs-id1169736657080\">Recognizing that [latex]{ \\cos }^{2}x+{ \\sin }^{2}x=1,[\/latex] by the Pythagorean theorem, we now have<\/p>\r\n\r\n<div id=\"fs-id1169739190112\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\frac{1}{{ \\cos }^{2}x}.[\/latex]<\/div>\r\n<p id=\"fs-id1169739225322\">Finally, use the identity [latex] \\sec x=\\frac{1}{ \\cos x}[\/latex] to obtain<\/p>\r\n\r\n<div id=\"fs-id1169736656627\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)={ \\sec }^{2}x.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169739273066\" class=\"exercise\">\r\n<div id=\"fs-id1169739273068\" class=\"textbox\">\r\n<p id=\"fs-id1169739273070\">Find the derivative of [latex]f(x)= \\cot x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[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)=\\text{\u2212}{ \\csc }^{2}x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739300236\" class=\"commentary\">\r\n<h4>Hint<\/h4>\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<\/div>\r\n<\/div>\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 key-takeaways theorem\">\r\n<h3>Derivatives of [latex] \\tan x, \\cot x, \\sec x,[\/latex] and [latex] \\csc x[\/latex]<\/h3>\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\">[latex]\\frac{d}{dx}( \\tan x)={ \\sec }^{2}x[\/latex]<\/div>\r\n<div id=\"fs-id1169739301143\" class=\"equation\">[latex]\\phantom{\\rule{0.72em}{0ex}}\\frac{d}{dx}( \\cot x)=\\text{\u2212}{ \\csc }^{2}x[\/latex]<\/div>\r\n<div id=\"fs-id1169739301181\" class=\"equation\">[latex]\\frac{d}{dx}( \\sec x)= \\sec x \\tan x[\/latex]<\/div>\r\n<div id=\"fs-id1169736658480\" class=\"equation\">[latex]\\frac{d}{dx}( \\csc x)=\\text{\u2212} \\csc x \\cot \\mathrm{x.}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739111296\" class=\"textbox examples\">\r\n<h3>Finding the Equation of a Tangent Line<\/h3>\r\n<div id=\"fs-id1169739111298\" class=\"exercise\">\r\n<div id=\"fs-id1169739111300\" class=\"textbox\">\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=\\frac{\\text{\u03c0}}{4}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[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\">[latex]f(\\frac{\\pi }{4})= \\cot \\frac{\\pi }{4}=1.[\/latex]<\/div>\r\n<p id=\"fs-id1169736656804\">Thus the tangent line passes through the point [latex](\\frac{\\pi }{4},1).[\/latex] Next, find the slope by finding the derivative of [latex]f(x)= \\cot x[\/latex] and evaluating it at [latex]\\frac{\\pi }{4}\\text{:}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169736589231\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\text{\u2212}{ \\csc }^{2}x\\text{ and }{f}^{\\prime }(\\frac{\\pi }{4})=\\text{\u2212}{ \\csc }^{2}(\\frac{\\pi }{4})=-2.[\/latex]<\/div>\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\">[latex]y-1=-2(x-\\frac{\\pi }{4})[\/latex]<\/div>\r\n<p id=\"fs-id1169739336035\">or equivalently,<\/p>\r\n\r\n<div id=\"fs-id1169739336038\" class=\"equation unnumbered\">[latex]y=-2x+1+\\frac{\\pi }{2}.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736655845\" class=\"textbox examples\">\r\n<h3>Finding the Derivative of Trigonometric Functions<\/h3>\r\n<div id=\"fs-id1169736655848\" class=\"exercise\">\r\n<div id=\"fs-id1169736655850\" class=\"textbox\">\r\n<p id=\"fs-id1169736655855\">Find the derivative of [latex]f(x)= \\csc x+x \\tan x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[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\">[latex]{f}^{\\prime }(x)=\\frac{d}{dx}( \\csc x)+\\frac{d}{dx}(x \\tan x).[\/latex]<\/div>\r\n<p id=\"fs-id1169736610172\">In the first term, [latex]\\frac{d}{dx}( \\csc x)=\\text{\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\">[latex]\\frac{d}{dx}(x \\tan x)=(1)( \\tan x)+({ \\sec }^{2}x)(x).[\/latex]<\/div>\r\n<p id=\"fs-id1169739265934\">Therefore, we have<\/p>\r\n\r\n<div id=\"fs-id1169739265938\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\text{\u2212} \\csc x \\cot x+ \\tan x+x{ \\sec }^{2}x.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739188143\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169739188146\" class=\"exercise\">\r\n<div id=\"fs-id1169739188148\" class=\"textbox\">\r\n<p id=\"fs-id1169739188150\">Find the derivative of [latex]f(x)=2 \\tan x-3 \\cot x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[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\r\n<\/div>\r\n<div id=\"fs-id1169736662612\" class=\"commentary\">\r\n<h4>Hint<\/h4>\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<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736662626\" class=\"textbox exercises checkpoint\">\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n\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<\/div>\r\n<div class=\"textbox shaded\">[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\r\n<\/div>\r\n<div class=\"commentary\">\r\n<h4>Hint<\/h4>\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<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739303779\" class=\"bc-section section\">\r\n<h1>Higher-Order Derivatives<\/h1>\r\n<p id=\"fs-id1169739303785\">The higher-order derivatives of [latex] \\sin x[\/latex] and [latex] \\cos x[\/latex] follow a repeating pattern. By following the pattern, we can find any higher-order derivative of [latex] \\sin x[\/latex] and [latex] \\cos x.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169739303827\" class=\"textbox examples\">\r\n<h3>Finding Higher-Order Derivatives of [latex]y= \\sin x[\/latex]<\/h3>\r\n<div id=\"fs-id1169739303829\" class=\"exercise\">\r\n<div id=\"fs-id1169739303832\" class=\"textbox\">\r\n<p id=\"fs-id1169739303850\">Find the first four derivatives of [latex]y= \\sin x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739303871\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739303871\"]\r\n<p id=\"fs-id1169739303871\">Each step in the chain is straightforward:<\/p>\r\n\r\n<div id=\"fs-id1169739303874\" class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; \\sin x\\hfill \\\\ \\hfill \\frac{dy}{dx}&amp; =\\hfill &amp; \\cos x\\hfill \\\\ \\hfill \\frac{{d}^{2}y}{d{x}^{2}}&amp; =\\hfill &amp; \\text{\u2212} \\sin x\\hfill \\\\ \\hfill \\frac{{d}^{3}y}{d{x}^{3}}&amp; =\\hfill &amp; \\text{\u2212} \\cos x\\hfill \\\\ \\hfill \\frac{{d}^{4}y}{d{x}^{4}}&amp; =\\hfill &amp; \\sin x.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739284979\" class=\"commentary\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1169739284984\">Once we recognize the pattern of derivatives, we can find any higher-order derivative by determining the step in the pattern to which it corresponds. For example, every fourth derivative of sin [latex]x[\/latex] equals sin [latex]x[\/latex], so<\/p>\r\n\r\n<div id=\"fs-id1169739284998\" class=\"equation unnumbered\">[latex]\\begin{array}{l}\\frac{{d}^{4}}{d{x}^{4}}( \\sin x)=\\frac{{d}^{8}}{d{x}^{8}}( \\sin x)=\\frac{{d}^{12}}{d{x}^{12}}( \\sin x)=\\text{\u2026}=\\frac{{d}^{4n}}{d{x}^{4n}}( \\sin x)= \\sin x\\\\ \\frac{{d}^{5}}{d{x}^{5}}( \\sin x)=\\frac{{d}^{9}}{d{x}^{9}}( \\sin x)=\\frac{{d}^{13}}{d{x}^{13}}( \\sin x)=\\text{\u2026}=\\frac{{d}^{4n+1}}{d{x}^{4n+1}}( \\sin x)= \\cos x.\\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739298009\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169739298012\" class=\"exercise\">\r\n<div id=\"fs-id1169739298015\" class=\"textbox\">\r\n<p id=\"fs-id1169739298017\">For [latex]y= \\cos x,[\/latex] find [latex]\\frac{{d}^{4}y}{d{x}^{4}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739298062\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739298062\"]\r\n<p id=\"fs-id1169739298062\">[latex] \\cos x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739298073\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169739298080\">See the previous example.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739298086\" class=\"textbox examples\">\r\n<h3>Using the Pattern for Higher-Order Derivatives of [latex]y= \\sin x[\/latex]<\/h3>\r\n<div id=\"fs-id1169739298088\" class=\"exercise\">\r\n<div id=\"fs-id1169739298090\" class=\"textbox\">\r\n<p id=\"fs-id1169739293614\">Find [latex]\\frac{{d}^{74}}{d{x}^{74}}( \\sin x).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739293655\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739293655\"]\r\n<p id=\"fs-id1169739293655\">We can see right away that for the 74th derivative of [latex] \\sin x,74=4(18)+2,[\/latex] so<\/p>\r\n\r\n<div id=\"fs-id1169739293692\" class=\"equation unnumbered\">[latex]\\frac{{d}^{74}}{d{x}^{74}}( \\sin x)=\\frac{{d}^{72+2}}{d{x}^{72+2}}( \\sin x)=\\frac{{d}^{2}}{d{x}^{2}}( \\sin x)=\\text{\u2212} \\sin x.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736595960\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169736595964\" class=\"exercise\">\r\n<div id=\"fs-id1169736595966\" class=\"textbox\">\r\n<p id=\"fs-id1169736595968\">For [latex]y= \\sin x,[\/latex] find [latex]\\frac{{d}^{59}}{d{x}^{59}}( \\sin x).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736596026\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736596026\"]\r\n<p id=\"fs-id1169736596026\">[latex]\\text{\u2212} \\cos x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736596039\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169736596046\">[latex]\\frac{{d}^{59}}{d{x}^{59}}( \\sin x)=\\frac{{d}^{4\u00b714+3}}{d{x}^{4\u00b714+3}}( \\sin x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739376125\" class=\"textbox examples\">\r\n<h3>An Application to Acceleration<\/h3>\r\n<div id=\"fs-id1169739376127\" class=\"exercise\">\r\n<div id=\"fs-id1169739376129\" class=\"textbox\">\r\n<p id=\"fs-id1169739376135\">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.[\/latex] Find [latex]v(\\pi \\text{\/}4)[\/latex] and [latex]a(\\pi \\text{\/}4).[\/latex] Compare these values and decide whether the particle is speeding up or slowing down.<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<p id=\"fs-id1169739376212\">First find [latex]v(t)={s}^{\\prime }(t)\\text{:}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1169739345785\" class=\"equation unnumbered\">[latex]v(t)={s}^{\\prime }(t)=\\text{\u2212} \\cos t.[\/latex]<\/div>\r\n<p id=\"fs-id1169739345829\">Thus,<\/p>\r\n\r\n<div id=\"fs-id1169739345832\" class=\"equation unnumbered\">[latex]v(\\frac{\\pi }{4})=-\\frac{1}{\\sqrt{2}}.[\/latex]<\/div>\r\n<p id=\"fs-id1169739345866\">Next, find [latex]a(t)={v}^{\\prime }(t).[\/latex] Thus, [latex]a(t)={v}^{\\prime }(t)= \\sin t[\/latex] and we have<\/p>\r\n\r\n<div id=\"fs-id1169739345937\" class=\"equation unnumbered\">[latex]a(\\frac{\\pi }{4})=\\frac{1}{\\sqrt{2}}.[\/latex]<\/div>\r\n<p id=\"fs-id1169739273581\">Since [latex]v(\\frac{\\pi }{4})=-\\frac{1}{\\sqrt{2}}&lt;0[\/latex] and [latex]a(\\frac{\\pi }{4})=\\frac{1}{\\sqrt{2}}&gt;0,[\/latex] we see that velocity and acceleration are acting in opposite directions; that is, the object is being accelerated in the direction opposite to the direction in which it is travelling. Consequently, the particle is slowing down.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739273655\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1169739273659\" class=\"exercise\">\r\n<div id=\"fs-id1169739273661\" class=\"textbox\">\r\n<p id=\"fs-id1169739273663\">A block attached to a spring is moving vertically. Its position at time [latex]t[\/latex] is given by [latex]s(t)=2 \\sin t.[\/latex] Find [latex]v(\\frac{5\\pi }{6})[\/latex] and [latex]a(\\frac{5\\pi }{6}).[\/latex] Compare these values and decide whether the block is speeding up or slowing down.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739325513\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739325513\"]\r\n<p id=\"fs-id1169739325513\">[latex]v(\\frac{5\\pi }{6})=\\text{\u2212}\\sqrt{3}&lt;0[\/latex] and [latex]a(\\frac{5\\pi }{6})=-1&lt;0.[\/latex] The block is speeding up.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739325579\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169739325585\">Use <a class=\"autogenerated-content\" href=\"#fs-id1169739376125\">(Figure)<\/a> as a guide.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739325596\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1169739325602\">\r\n \t<li>We can find the derivatives of sin [latex]x[\/latex] and cos [latex]x[\/latex] by using the definition of derivative and the limit formulas found earlier. The results are\r\n<div id=\"fs-id1169736589261\" class=\"equation unnumbered\">[latex]\\frac{d}{dx} \\sin x= \\cos x\\frac{d}{dx} \\cos x=\\text{\u2212} \\sin x.[\/latex]<\/div><\/li>\r\n \t<li>With these two formulas, we can determine the derivatives of all six basic trigonometric functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169736603422\" class=\"key-equations\">\r\n<h1>Key Equations<\/h1>\r\n<ul id=\"fs-id1169736603429\">\r\n \t<li><strong>Derivative of sine function<\/strong>\r\n[latex]\\frac{d}{dx}( \\sin x)= \\cos x[\/latex]<\/li>\r\n \t<li><strong>Derivative of cosine function<\/strong>\r\n[latex]\\frac{d}{dx}( \\cos x)=\\text{\u2212} \\sin x[\/latex]<\/li>\r\n \t<li><strong>Derivative of tangent function<\/strong>\r\n[latex]\\frac{d}{dx}( \\tan x)={ \\sec }^{2}x[\/latex]<\/li>\r\n \t<li><strong>Derivative of cotangent function<\/strong>\r\n[latex]\\frac{d}{dx}( \\cot x)=\\text{\u2212}{ \\csc }^{2}x[\/latex]<\/li>\r\n \t<li><strong>Derivative of secant function<\/strong>\r\n[latex]\\frac{d}{dx}( \\sec x)= \\sec x \\tan x[\/latex]<\/li>\r\n \t<li><strong>Derivative of cosecant function<\/strong>\r\n[latex]\\frac{d}{dx}( \\csc x)=\\text{\u2212} \\csc x \\cot x[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169736597627\" class=\"textbox exercises\">\r\n<p id=\"fs-id1169736597631\">For the following exercises, find [latex]\\frac{dy}{dx}[\/latex] for the given functions.<\/p>\r\n\r\n<div id=\"fs-id1169736597649\" class=\"exercise\">\r\n<div id=\"fs-id1169736597652\" class=\"textbox\">\r\n<p id=\"fs-id1169736597654\">[latex]y={x}^{2}- \\sec x+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736597682\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736597682\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736597682\"][latex]\\frac{dy}{dx}=2x- \\sec x \\tan x[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736597729\" class=\"exercise\">\r\n<div id=\"fs-id1169736597731\" class=\"textbox\">\r\n<p id=\"fs-id1169736597733\">[latex]y=3 \\csc x+\\frac{5}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736658534\" class=\"exercise\">\r\n<div id=\"fs-id1169736658536\" class=\"textbox\">\r\n<p id=\"fs-id1169736658538\">[latex]y={x}^{2} \\cot x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736658563\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736658563\"]\r\n<p id=\"fs-id1169736658563\">[latex]\\frac{dy}{dx}=2x \\cot x-{x}^{2}{ \\csc }^{2}x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736658614\" class=\"exercise\">\r\n<div id=\"fs-id1169736658616\" class=\"textbox\">\r\n<p id=\"fs-id1169736658618\">[latex]y=x-{x}^{3} \\sin x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739275164\" class=\"exercise\">\r\n<div id=\"fs-id1169739275166\" class=\"textbox\">\r\n<p id=\"fs-id1169739275168\">[latex]y=\\frac{ \\sec x}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739275191\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739275191\"]\r\n<p id=\"fs-id1169739275191\">[latex]\\frac{dy}{dx}=\\frac{x \\sec x \\tan x- \\sec x}{{x}^{2}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739275253\" class=\"exercise\">\r\n<div id=\"fs-id1169739275255\" class=\"textbox\">\r\n<p id=\"fs-id1169739275257\">[latex]y= \\sin x \\tan x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739303603\" class=\"exercise\">\r\n<div id=\"fs-id1169739303605\" class=\"textbox\">\r\n<p id=\"fs-id1169739303607\">[latex]y=(x+ \\cos x)(1- \\sin x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739303655\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739303655\"]\r\n<p id=\"fs-id1169739303655\">[latex]\\frac{dy}{dx}=(1- \\sin x)(1- \\sin x)- \\cos x(x+ \\cos x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739303738\" class=\"exercise\">\r\n<div id=\"fs-id1169739303741\" class=\"textbox\">\r\n<p id=\"fs-id1169739303743\">[latex]y=\\frac{ \\tan x}{1- \\sec x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739350812\" class=\"exercise\">\r\n<div id=\"fs-id1169739350814\" class=\"textbox\">\r\n<p id=\"fs-id1169739350817\">[latex]y=\\frac{1- \\cot x}{1+ \\cot x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739350855\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739350855\"]\r\n<p id=\"fs-id1169739350855\">[latex]\\frac{dy}{dx}=\\frac{2{ \\csc }^{2}x}{{(1+ \\cot x)}^{2}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736593529\" class=\"exercise\">\r\n<div id=\"fs-id1169736593531\" class=\"textbox\">\r\n<p id=\"fs-id1169736593533\">[latex]y= \\cos x(1+ \\csc x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169736593662\">For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of [latex]x.[\/latex] Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.<\/p>\r\n\r\n<div id=\"fs-id1169736593674\" class=\"exercise\">\r\n<div id=\"fs-id1169739266595\" class=\"textbox\">\r\n<p id=\"fs-id1169739266597\"><strong>[T]<\/strong>[latex]f(x)=\\text{\u2212} \\sin x,x=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739266640\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739266640\"]\r\n<p id=\"fs-id1169739266640\">[latex]y=\\text{\u2212}x[\/latex]<\/p>\r\n<span id=\"fs-id1169739266655\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205425\/CNX_Calc_Figure_03_05_201.jpg\" alt=\"The graph shows negative sin(x) and the straight line T(x) with slope \u22121 and y intercept 0.\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739266669\" class=\"exercise\">\r\n<div id=\"fs-id1169739266671\" class=\"textbox\">\r\n<p id=\"fs-id1169739266673\"><strong>[T]<\/strong>[latex]f(x)= \\csc x,x=\\frac{\\pi }{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739266742\" class=\"exercise\">\r\n<div id=\"fs-id1169739266744\" class=\"textbox\">\r\n<p id=\"fs-id1169739266746\"><strong>[T]<\/strong>[latex]f(x)=1+ \\cos x,x=\\frac{3\\pi }{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736655158\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736655158\"]\r\n<p id=\"fs-id1169736655158\">[latex]y=x+\\frac{2-3\\pi }{2}[\/latex]<\/p>\r\n<span id=\"fs-id1169736655187\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205428\/CNX_Calc_Figure_03_05_203.jpg\" alt=\"The graph shows the cosine function shifted up one and has the straight line T(x) with slope 1 and y intercept (2 \u2013 3\u03c0)\/2.\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736655200\" class=\"exercise\">\r\n<div id=\"fs-id1169736655202\" class=\"textbox\">\r\n<p id=\"fs-id1169736655204\"><strong>[T]<\/strong>[latex]f(x)= \\sec x,x=\\frac{\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736655301\" class=\"exercise\">\r\n<div id=\"fs-id1169736655303\" class=\"textbox\">\r\n<p id=\"fs-id1169736655305\"><strong>[T]<\/strong>[latex]f(x)={x}^{2}- \\tan xx=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\n[latex]y=\\text{\u2212}x[\/latex]\r\n\r\n<span id=\"fs-id1169739305478\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205430\/CNX_Calc_Figure_03_05_205.jpg\" alt=\"The graph shows the function as starting at (\u22121, 3), decreasing to the origin, continuing to slowly decrease to about (1, \u22120.5), at which point it decreases very quickly.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739305492\" class=\"exercise\">\r\n<div id=\"fs-id1169739305494\" class=\"textbox\">\r\n<p id=\"fs-id1169739305496\"><strong>[T]<\/strong>[latex]f(x)=5 \\cot xx=\\frac{\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169739305590\">For the following exercises, find [latex]\\frac{{d}^{2}y}{d{x}^{2}}[\/latex] for the given functions.<\/p>\r\n\r\n<div id=\"fs-id1169739305614\" class=\"exercise\">\r\n<div id=\"fs-id1169739305617\" class=\"textbox\">\r\n<p id=\"fs-id1169739305619\">[latex]y=x \\sin x- \\cos x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736662274\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736662274\"]\r\n<p id=\"fs-id1169736662274\">[latex]3 \\cos x-x \\sin x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736662308\" class=\"exercise\">\r\n<div id=\"fs-id1169736662310\" class=\"textbox\">\r\n\r\n[latex]y= \\sin x \\cos x[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736662371\" class=\"exercise\">\r\n<div id=\"fs-id1169736662373\" class=\"textbox\">\r\n<p id=\"fs-id1169736662375\">[latex]y=x-\\frac{1}{2} \\sin x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736662404\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736662404\"]\r\n<p id=\"fs-id1169736662404\">[latex]\\frac{1}{2} \\sin x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736662423\" class=\"exercise\">\r\n<div id=\"fs-id1169736662425\" class=\"textbox\">\r\n<p id=\"fs-id1169736662427\">[latex]y=\\frac{1}{x}+ \\tan x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739303429\" class=\"exercise\">\r\n<div id=\"fs-id1169739303431\" class=\"textbox\">\r\n<p id=\"fs-id1169739303433\">[latex]y=2 \\csc x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739303458\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739303458\"]\r\n<p id=\"fs-id1169739303458\">[latex] \\csc (x)(3{ \\csc }^{2}(x)-1+{ \\cot }^{2}(x))[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739303516\" class=\"exercise\">\r\n<div id=\"fs-id1169739303518\" class=\"textbox\">\r\n<p id=\"fs-id1169739303520\">[latex]y={ \\sec }^{2}x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739335809\" class=\"exercise\">\r\n<div id=\"fs-id1169739335812\" class=\"textbox\">\r\n<p id=\"fs-id1169739335814\">Find all [latex]x[\/latex] values on the graph of [latex]f(x)=-3 \\sin x \\cos x[\/latex] where the tangent line is horizontal.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739335862\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739335862\"]\r\n<p id=\"fs-id1169739335862\">[latex]\\frac{(2n+1)\\pi }{4},\\text{ where }n\\text{is an integer}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739335906\" class=\"exercise\">\r\n<div id=\"fs-id1169739335909\" class=\"textbox\">\r\n<p id=\"fs-id1169739335911\">Find all [latex]x[\/latex] values on the graph of [latex]f(x)=x-2 \\cos x[\/latex] for [latex]0&lt;x&lt;2\\pi [\/latex] where the tangent line has slope 2.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739335992\" class=\"exercise\">\r\n<div id=\"fs-id1169739335994\" class=\"textbox\">\r\n<p id=\"fs-id1169739335996\">Let [latex]f(x)= \\cot x.[\/latex] Determine the points on the graph of [latex]f[\/latex] for [latex]0&lt;x&lt;2\\pi [\/latex] where the tangent line(s) is (are) parallel to the line [latex]y=-2x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736613904\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736613904\"]\r\n<p id=\"fs-id1169736613904\">[latex](\\frac{\\pi }{4},1),(\\frac{3\\pi }{4},-1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736613947\" class=\"exercise\">\r\n<div id=\"fs-id1169736613949\" class=\"textbox\">\r\n\r\n<strong>[T]<\/strong> A mass on a spring bounces up and down in simple harmonic motion, modeled by the function [latex]s(t)=-6 \\cos t[\/latex] where [latex]s[\/latex] is measured in inches and [latex]t[\/latex] is measured in seconds. Find the rate at which the spring is oscillating at [latex]t=5[\/latex] s.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736614037\" class=\"exercise\">\r\n<div id=\"fs-id1169736614039\" class=\"textbox\">\r\n\r\nLet the position of a swinging pendulum in simple harmonic motion be given by [latex]s(t)=a \\cos t+b \\sin t.[\/latex] Find the constants [latex]a[\/latex] and [latex]b[\/latex] such that when the velocity is 3 cm\/s, [latex]s=0[\/latex] and [latex]t=0.[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739341388\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739341388\"]\r\n<p id=\"fs-id1169739341388\">[latex]a=0,b=3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739341409\" class=\"exercise\">\r\n<div id=\"fs-id1169739341411\" class=\"textbox\">\r\n<p id=\"fs-id1169739341414\">After a diver jumps off a diving board, the edge of the board oscillates with position given by [latex]s(t)=-5 \\cos t[\/latex] cm at [latex]t[\/latex] seconds after the jump.<\/p>\r\n\r\n<ol id=\"fs-id1169739341452\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Sketch one period of the position function for [latex]t\\ge 0.[\/latex]<\/li>\r\n \t<li>Find the velocity function.<\/li>\r\n \t<li>Sketch one period of the velocity function for [latex]t\\ge 0.[\/latex]<\/li>\r\n \t<li>Determine the times when the velocity is 0 over one period.<\/li>\r\n \t<li>Find the acceleration function.<\/li>\r\n \t<li>Sketch one period of the acceleration function for [latex]t\\ge 0.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739353376\" class=\"exercise\">\r\n<div id=\"fs-id1169739353378\" class=\"textbox\">\r\n<p id=\"fs-id1169739353380\">The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by [latex]y=10+5 \\sin x[\/latex] where [latex]y[\/latex] is the number of hamburgers sold and [latex]x[\/latex] represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find [latex]y\\prime [\/latex] and determine the intervals where the number of burgers being sold is increasing.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739353429\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739353429\"]\r\n<p id=\"fs-id1169739353429\">[latex]{y}^{\\prime }=5 \\cos (x),[\/latex] increasing on [latex](0,\\frac{\\pi }{2}),(\\frac{3\\pi }{2},\\frac{5\\pi }{2}),[\/latex] and [latex](\\frac{7\\pi }{2},12)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739307888\" class=\"exercise\">\r\n<div id=\"fs-id1169739307890\" class=\"textbox\">\r\n<p id=\"fs-id1169739307892\"><strong>[T]<\/strong> The amount of rainfall per month in Phoenix, Arizona, can be approximated by [latex]y(t)=0.5+0.3 \\cos t,[\/latex] where [latex]t[\/latex] is months since January. Find [latex]{y}^{\\prime }[\/latex] and use a calculator to determine the intervals where the amount of rain falling is decreasing.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169739264125\">For the following exercises, use the quotient rule to derive the given equations.<\/p>\r\n\r\n<div id=\"fs-id1169739264128\" class=\"exercise\">\r\n<div id=\"fs-id1169739264130\" class=\"textbox\">\r\n<p id=\"fs-id1169739264132\">[latex]\\frac{d}{dx}( \\cot x)=\\text{\u2212}{ \\csc }^{2}x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739264176\" class=\"exercise\">\r\n<div id=\"fs-id1169739264178\" class=\"textbox\">\r\n<p id=\"fs-id1169739264180\">[latex]\\frac{d}{dx}( \\sec x)= \\sec x \\tan x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739264230\" class=\"exercise\">\r\n<div id=\"fs-id1169739264233\" class=\"textbox\">\r\n<p id=\"fs-id1169739264235\">[latex]\\frac{d}{dx}( \\csc x)=\\text{\u2212} \\csc x \\cot x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739264287\" class=\"exercise\">\r\n<div id=\"fs-id1169739264289\" class=\"textbox\">\r\n<p id=\"fs-id1169739264292\">Use the definition of derivative and the identity<\/p>\r\n<p id=\"fs-id1169739264295\">[latex] \\cos (x+h)= \\cos x \\cos h- \\sin x \\sin h[\/latex] to prove that [latex]\\frac{d( \\cos x)}{dx}=\\text{\u2212} \\sin x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169739289336\">For the following exercises, find the requested higher-order derivative for the given functions.<\/p>\r\n\r\n<div id=\"fs-id1169739289340\" class=\"exercise\">\r\n<div id=\"fs-id1169739289343\" class=\"textbox\">\r\n<p id=\"fs-id1169739289345\">[latex]\\frac{{d}^{3}y}{d{x}^{3}}[\/latex] of [latex]y=3 \\cos x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<p id=\"fs-id1169739289390\">[latex]3 \\sin x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739289409\" class=\"exercise\">\r\n<div id=\"fs-id1169739289411\" class=\"textbox\">\r\n<p id=\"fs-id1169739289414\">[latex]\\frac{{d}^{2}y}{d{x}^{2}}[\/latex] of [latex]y=3 \\sin x+{x}^{2} \\cos x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1169736592469\" class=\"textbox\">\r\n<p id=\"fs-id1169736592471\">[latex]\\frac{{d}^{4}y}{d{x}^{4}}[\/latex] of [latex]y=5 \\cos x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169736592517\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736592517\"]\r\n<p id=\"fs-id1169736592517\">[latex]5 \\cos x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736592536\" class=\"exercise\">\r\n<div id=\"fs-id1169736592538\" class=\"textbox\">\r\n<p id=\"fs-id1169736592540\">[latex]\\frac{{d}^{2}y}{d{x}^{2}}[\/latex] of [latex]y= \\sec x+ \\cot x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739301937\" class=\"exercise\">\r\n<div id=\"fs-id1169739301939\" class=\"textbox\">\r\n<p id=\"fs-id1169739301941\">[latex]\\frac{{d}^{3}y}{d{x}^{3}}[\/latex] of [latex]y={x}^{10}- \\sec x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1169739301990\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739301990\"]\r\n<p id=\"fs-id1169739301990\">[latex]720{x}^{7}-5 \\tan (x){ \\sec }^{3}(x)-{ \\tan }^{3}(x) \\sec (x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/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<p>One of the most important types of motion in physics is simple harmonic motion, which is associated with such systems as an object with mass oscillating on a spring. Simple harmonic motion can be described by using either sine or cosine functions. In this section we expand our knowledge of derivative formulas to include derivatives of these and other trigonometric functions. We begin with the derivatives of the sine and cosine functions and then use them to obtain formulas for the derivatives of the remaining four trigonometric functions. Being able to calculate the derivatives of the sine and cosine functions will enable us to find the velocity and acceleration of simple harmonic motion.<\/p>\n<div id=\"fs-id1169739300487\" class=\"bc-section section\">\n<h1>Derivatives of the Sine and Cosine Functions<\/h1>\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\">[latex]{f}^{\\prime }(x)=\\underset{h\\to 0}{\\text{lim}}\\frac{f(x+h)-f(x)}{h}.[\/latex]<\/div>\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\">[latex]\\frac{d}{dx}( \\sin x)\\approx \\frac{ \\sin (x+0.01)- \\sin x}{0.01}[\/latex]<\/div>\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] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_03_05_001\">(Figure)<\/a>).<\/p>\n<div id=\"CNX_Calc_Figure_03_05_001\" class=\"wp-caption aligncenter\">\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\"><strong>Figure 1.<\/strong> The graph of the function [latex]D(x)[\/latex] looks a lot like a cosine curve.<\/p>\n<\/div>\n<\/div>\n<div class=\"wp-caption-text\"><\/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\">[latex]\\frac{d}{dx}( \\sin x)= \\cos x.[\/latex]<\/div>\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\">[latex]\\frac{d}{dx}( \\cos x)=\\text{\u2212} \\sin \\mathrm{x.}[\/latex]<\/div>\n<div id=\"fs-id1169739098813\" class=\"textbox key-takeaways theorem\">\n<h3>The Derivatives of sin [latex]x[\/latex] and cos [latex]x[\/latex]<\/h3>\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\">[latex]\\frac{d}{dx}( \\sin x)= \\cos x[\/latex]<\/div>\n<div class=\"equation\">[latex]\\frac{d}{dx}( \\cos x)=\\text{\u2212} \\sin x[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1169738980720\" class=\"bc-section section\">\n<h2>Proof<\/h2>\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)=\\text{\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 <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction-2\/\">Introduction to Limits<\/a>:<\/p>\n<div id=\"fs-id1169738960853\" class=\"equation unnumbered\">[latex]\\underset{h\\to 0}{\\text{lim}}\\frac{ \\sin h}{h}=1\\text{ and }\\underset{h\\to 0}{\\text{lim}}\\frac{\\text{cosh}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 <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_03_05_002\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Calc_Figure_03_05_002\" class=\"wp-caption aligncenter\">\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<\/div>\n<div class=\"wp-caption-text\"><\/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\">[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\">[latex]\\begin{array}{ccccc}\\hfill \\frac{d}{dx} \\sin x& =\\underset{h\\to 0}{\\text{lim}}\\frac{ \\sin (x+h)- \\sin x}{h}\\hfill & & & \\text{Apply the definition of the derivative.}\\hfill \\\\ & =\\underset{h\\to 0}{\\text{lim}}\\frac{ \\sin x \\cos h+ \\cos x \\sin h- \\sin x}{h}\\hfill & & & \\text{Use trig identity for the sine of the sum of two angles.}\\hfill \\\\ & =\\underset{h\\to 0}{\\text{lim}}(\\frac{ \\sin x \\cos h- \\sin x}{h}+\\frac{ \\cos x \\sin h}{h})\\hfill & & & \\text{Regroup.}\\hfill \\\\ & =\\underset{h\\to 0}{\\text{lim}}( \\sin x(\\frac{ \\cos xh-1}{h})+ \\cos x(\\frac{ \\sin h}{h}))\\hfill & & & \\text{Factor out} \\sin x\\text{ and } \\cos x.\\hfill \\\\ & = \\sin x(0)+ \\cos x(1)\\hfill & & & \\text{Apply trig limit formulas.}\\hfill \\\\ & = \\cos x\\hfill & & & \\text{Simplify.}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1169739009931\">\u25a1<\/p>\n<p id=\"fs-id1169739186572\"><a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_03_05_003\">(Figure)<\/a> 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 id=\"CNX_Calc_Figure_03_05_003\" class=\"wp-caption aligncenter\">\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>\n<div class=\"wp-caption-text\"><\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1169738907624\" class=\"exercise\">\n<div id=\"fs-id1169739015112\" class=\"textbox\">\n<h3>Differentiating a Function Containing sin [latex]x[\/latex]<\/h3>\n<p id=\"fs-id1169739269454\">Find the derivative of [latex]f(x)=5{x}^{3} \\sin x.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169739028319\" class=\"textbox shaded\">\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\">[latex]\\begin{array}{cc}\\hfill f\\prime (x)& =\\frac{d}{dx}(5{x}^{3})\u00b7 \\sin x+\\frac{d}{dx}( \\sin x)\u00b75{x}^{3}\\hfill \\\\ & =15{x}^{2}\u00b7 \\sin x+ \\cos x\u00b75{x}^{3}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1169739036358\">After simplifying, we obtain<\/p>\n<div id=\"fs-id1169739269866\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=15{x}^{2} \\sin x+5{x}^{3} \\cos x.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739304325\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169739327353\" class=\"exercise\">\n<div id=\"fs-id1169739009945\" class=\"textbox\">\n<p id=\"fs-id1169736660715\">Find the derivative of [latex]f(x)= \\sin x \\cos x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\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 id=\"fs-id1169736614177\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169738821957\">Don\u2019t forget to use the product rule.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738889578\" class=\"textbox examples\">\n<h3>Finding the Derivative of a Function Containing cos [latex]x[\/latex]<\/h3>\n<div id=\"fs-id1169736615213\" class=\"exercise\">\n<div id=\"fs-id1169738994002\" class=\"textbox\">\n<p id=\"fs-id1169739229519\">Find the derivative of [latex]g(x)=\\frac{ \\cos x}{4{x}^{2}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\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\">[latex]{g}^{\\prime }(x)=\\frac{(\\text{\u2212} \\sin x)4{x}^{2}-8x( \\cos x)}{{(4{x}^{2})}^{2}}.[\/latex]<\/div>\n<p id=\"fs-id1169738949562\">Simplifying, we obtain<\/p>\n<div id=\"fs-id1169739179211\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill {g}^{\\prime }(x)& =\\frac{-4{x}^{2} \\sin x-8x \\cos x}{16{x}^{4}}\\hfill \\\\ & =\\frac{\\text{\u2212}x \\sin x-2 \\cos x}{4{x}^{3}}.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736615168\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169738971411\" class=\"exercise\">\n<div id=\"fs-id1169738971413\" class=\"textbox\">\n<p id=\"fs-id1169738971415\">Find the derivative of [latex]f(x)=\\frac{x}{ \\cos x}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\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 id=\"fs-id1169736654818\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169736587923\">Use the quotient rule.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739220928\" class=\"textbox examples\">\n<h3>An Application to Velocity<\/h3>\n<div id=\"fs-id1169739220930\" class=\"exercise\">\n<div id=\"fs-id1169739220933\" class=\"textbox\">\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>\n<div class=\"textbox shaded\">\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\">[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\">[latex]2 \\cos t-1=0\\text{ for }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] 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>\n<\/div>\n<div id=\"fs-id1169739327874\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169739327877\" class=\"exercise\">\n<div id=\"fs-id1169738824891\" class=\"textbox\">\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>\n<div class=\"textbox shaded\">\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<\/div>\n<div id=\"fs-id1169739189957\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169739189964\">Use the previous example as a guide.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736662797\" class=\"bc-section section\">\n<h1>Derivatives of Other Trigonometric Functions<\/h1>\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=\"textbox examples\">\n<h3>The Derivative of the Tangent Function<\/h3>\n<div id=\"fs-id1169736589201\" class=\"exercise\">\n<div id=\"fs-id1169736589203\" class=\"textbox\">\n<p id=\"fs-id1169739303221\">Find the derivative of [latex]f(x)= \\tan x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\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\">[latex]f(x)= \\tan x=\\frac{ \\sin x}{ \\cos x}.[\/latex]<\/div>\n<p id=\"fs-id1169736615154\">Now apply the quotient rule to obtain<\/p>\n<div id=\"fs-id1169736615157\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\frac{ \\cos x \\cos x-(\\text{\u2212} \\sin x) \\sin x}{{( \\cos x)}^{2}}.[\/latex]<\/div>\n<p id=\"fs-id1169739111151\">Simplifying, we obtain<\/p>\n<div id=\"fs-id1169739111154\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\frac{{ \\cos }^{2}x+{ \\sin }^{2}x}{{ \\cos }^{2}x}.[\/latex]<\/div>\n<p id=\"fs-id1169736657080\">Recognizing that [latex]{ \\cos }^{2}x+{ \\sin }^{2}x=1,[\/latex] by the Pythagorean theorem, we now have<\/p>\n<div id=\"fs-id1169739190112\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\frac{1}{{ \\cos }^{2}x}.[\/latex]<\/div>\n<p id=\"fs-id1169739225322\">Finally, use the identity [latex]\\sec x=\\frac{1}{ \\cos x}[\/latex] to obtain<\/p>\n<div id=\"fs-id1169736656627\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)={ \\sec }^{2}x.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169739273066\" class=\"exercise\">\n<div id=\"fs-id1169739273068\" class=\"textbox\">\n<p id=\"fs-id1169739273070\">Find the derivative of [latex]f(x)= \\cot x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\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)=\\text{\u2212}{ \\csc }^{2}x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169739300236\" class=\"commentary\">\n<h4>Hint<\/h4>\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>\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 key-takeaways theorem\">\n<h3>Derivatives of [latex]\\tan x, \\cot x, \\sec x,[\/latex] and [latex]\\csc x[\/latex]<\/h3>\n<p id=\"fs-id1169739299818\">The derivatives of the remaining trigonometric functions are as follows:<\/p>\n<div id=\"fs-id1169739299822\" class=\"equation\">[latex]\\frac{d}{dx}( \\tan x)={ \\sec }^{2}x[\/latex]<\/div>\n<div id=\"fs-id1169739301143\" class=\"equation\">[latex]\\phantom{\\rule{0.72em}{0ex}}\\frac{d}{dx}( \\cot x)=\\text{\u2212}{ \\csc }^{2}x[\/latex]<\/div>\n<div id=\"fs-id1169739301181\" class=\"equation\">[latex]\\frac{d}{dx}( \\sec x)= \\sec x \\tan x[\/latex]<\/div>\n<div id=\"fs-id1169736658480\" class=\"equation\">[latex]\\frac{d}{dx}( \\csc x)=\\text{\u2212} \\csc x \\cot \\mathrm{x.}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1169739111296\" class=\"textbox examples\">\n<h3>Finding the Equation of a Tangent Line<\/h3>\n<div id=\"fs-id1169739111298\" class=\"exercise\">\n<div id=\"fs-id1169739111300\" class=\"textbox\">\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=\\frac{\\text{\u03c0}}{4}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\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\">[latex]f(\\frac{\\pi }{4})= \\cot \\frac{\\pi }{4}=1.[\/latex]<\/div>\n<p id=\"fs-id1169736656804\">Thus the tangent line passes through the point [latex](\\frac{\\pi }{4},1).[\/latex] Next, find the slope by finding the derivative of [latex]f(x)= \\cot x[\/latex] and evaluating it at [latex]\\frac{\\pi }{4}\\text{:}[\/latex]<\/p>\n<div id=\"fs-id1169736589231\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\text{\u2212}{ \\csc }^{2}x\\text{ and }{f}^{\\prime }(\\frac{\\pi }{4})=\\text{\u2212}{ \\csc }^{2}(\\frac{\\pi }{4})=-2.[\/latex]<\/div>\n<p id=\"fs-id1169739111215\">Using the point-slope equation of the line, we obtain<\/p>\n<div id=\"fs-id1169739111218\" class=\"equation unnumbered\">[latex]y-1=-2(x-\\frac{\\pi }{4})[\/latex]<\/div>\n<p id=\"fs-id1169739336035\">or equivalently,<\/p>\n<div id=\"fs-id1169739336038\" class=\"equation unnumbered\">[latex]y=-2x+1+\\frac{\\pi }{2}.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736655845\" class=\"textbox examples\">\n<h3>Finding the Derivative of Trigonometric Functions<\/h3>\n<div id=\"fs-id1169736655848\" class=\"exercise\">\n<div id=\"fs-id1169736655850\" class=\"textbox\">\n<p id=\"fs-id1169736655855\">Find the derivative of [latex]f(x)= \\csc x+x \\tan x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\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\">[latex]{f}^{\\prime }(x)=\\frac{d}{dx}( \\csc x)+\\frac{d}{dx}(x \\tan x).[\/latex]<\/div>\n<p id=\"fs-id1169736610172\">In the first term, [latex]\\frac{d}{dx}( \\csc x)=\\text{\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\">[latex]\\frac{d}{dx}(x \\tan x)=(1)( \\tan x)+({ \\sec }^{2}x)(x).[\/latex]<\/div>\n<p id=\"fs-id1169739265934\">Therefore, we have<\/p>\n<div id=\"fs-id1169739265938\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\text{\u2212} \\csc x \\cot x+ \\tan x+x{ \\sec }^{2}x.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739188143\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169739188146\" class=\"exercise\">\n<div id=\"fs-id1169739188148\" class=\"textbox\">\n<p id=\"fs-id1169739188150\">Find the derivative of [latex]f(x)=2 \\tan x-3 \\cot x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\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 id=\"fs-id1169736662612\" class=\"commentary\">\n<h4>Hint<\/h4>\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>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736662626\" class=\"textbox exercises checkpoint\">\n<div class=\"exercise\">\n<div class=\"textbox\">\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>\n<div class=\"textbox shaded\">\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 class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169739303758\">Evaluate the derivative at [latex]x=\\frac{\\pi }{6}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739303779\" class=\"bc-section section\">\n<h1>Higher-Order Derivatives<\/h1>\n<p id=\"fs-id1169739303785\">The higher-order derivatives of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex] follow a repeating pattern. By following the pattern, we can find any higher-order derivative of [latex]\\sin x[\/latex] and [latex]\\cos x.[\/latex]<\/p>\n<div id=\"fs-id1169739303827\" class=\"textbox examples\">\n<h3>Finding Higher-Order Derivatives of [latex]y= \\sin x[\/latex]<\/h3>\n<div id=\"fs-id1169739303829\" class=\"exercise\">\n<div id=\"fs-id1169739303832\" class=\"textbox\">\n<p id=\"fs-id1169739303850\">Find the first four derivatives of [latex]y= \\sin x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739303871\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739303871\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739303871\">Each step in the chain is straightforward:<\/p>\n<div id=\"fs-id1169739303874\" class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill y& =\\hfill & \\sin x\\hfill \\\\ \\hfill \\frac{dy}{dx}& =\\hfill & \\cos x\\hfill \\\\ \\hfill \\frac{{d}^{2}y}{d{x}^{2}}& =\\hfill & \\text{\u2212} \\sin x\\hfill \\\\ \\hfill \\frac{{d}^{3}y}{d{x}^{3}}& =\\hfill & \\text{\u2212} \\cos x\\hfill \\\\ \\hfill \\frac{{d}^{4}y}{d{x}^{4}}& =\\hfill & \\sin x.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739284979\" class=\"commentary\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1169739284984\">Once we recognize the pattern of derivatives, we can find any higher-order derivative by determining the step in the pattern to which it corresponds. For example, every fourth derivative of sin [latex]x[\/latex] equals sin [latex]x[\/latex], so<\/p>\n<div id=\"fs-id1169739284998\" class=\"equation unnumbered\">[latex]\\begin{array}{l}\\frac{{d}^{4}}{d{x}^{4}}( \\sin x)=\\frac{{d}^{8}}{d{x}^{8}}( \\sin x)=\\frac{{d}^{12}}{d{x}^{12}}( \\sin x)=\\text{\u2026}=\\frac{{d}^{4n}}{d{x}^{4n}}( \\sin x)= \\sin x\\\\ \\frac{{d}^{5}}{d{x}^{5}}( \\sin x)=\\frac{{d}^{9}}{d{x}^{9}}( \\sin x)=\\frac{{d}^{13}}{d{x}^{13}}( \\sin x)=\\text{\u2026}=\\frac{{d}^{4n+1}}{d{x}^{4n+1}}( \\sin x)= \\cos x.\\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739298009\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169739298012\" class=\"exercise\">\n<div id=\"fs-id1169739298015\" class=\"textbox\">\n<p id=\"fs-id1169739298017\">For [latex]y= \\cos x,[\/latex] find [latex]\\frac{{d}^{4}y}{d{x}^{4}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739298062\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739298062\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739298062\">[latex]\\cos x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169739298073\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169739298080\">See the previous example.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739298086\" class=\"textbox examples\">\n<h3>Using the Pattern for Higher-Order Derivatives of [latex]y= \\sin x[\/latex]<\/h3>\n<div id=\"fs-id1169739298088\" class=\"exercise\">\n<div id=\"fs-id1169739298090\" class=\"textbox\">\n<p id=\"fs-id1169739293614\">Find [latex]\\frac{{d}^{74}}{d{x}^{74}}( \\sin x).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739293655\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739293655\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739293655\">We can see right away that for the 74th derivative of [latex]\\sin x,74=4(18)+2,[\/latex] so<\/p>\n<div id=\"fs-id1169739293692\" class=\"equation unnumbered\">[latex]\\frac{{d}^{74}}{d{x}^{74}}( \\sin x)=\\frac{{d}^{72+2}}{d{x}^{72+2}}( \\sin x)=\\frac{{d}^{2}}{d{x}^{2}}( \\sin x)=\\text{\u2212} \\sin x.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736595960\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169736595964\" class=\"exercise\">\n<div id=\"fs-id1169736595966\" class=\"textbox\">\n<p id=\"fs-id1169736595968\">For [latex]y= \\sin x,[\/latex] find [latex]\\frac{{d}^{59}}{d{x}^{59}}( \\sin x).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736596026\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736596026\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736596026\">[latex]\\text{\u2212} \\cos x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169736596039\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169736596046\">[latex]\\frac{{d}^{59}}{d{x}^{59}}( \\sin x)=\\frac{{d}^{4\u00b714+3}}{d{x}^{4\u00b714+3}}( \\sin x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739376125\" class=\"textbox examples\">\n<h3>An Application to Acceleration<\/h3>\n<div id=\"fs-id1169739376127\" class=\"exercise\">\n<div id=\"fs-id1169739376129\" class=\"textbox\">\n<p id=\"fs-id1169739376135\">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.[\/latex] Find [latex]v(\\pi \\text{\/}4)[\/latex] and [latex]a(\\pi \\text{\/}4).[\/latex] Compare these values and decide whether the particle is speeding up or slowing down.<\/p>\n<\/div>\n<div class=\"solution\">\n<p id=\"fs-id1169739376212\">First find [latex]v(t)={s}^{\\prime }(t)\\text{:}[\/latex]<\/p>\n<div id=\"fs-id1169739345785\" class=\"equation unnumbered\">[latex]v(t)={s}^{\\prime }(t)=\\text{\u2212} \\cos t.[\/latex]<\/div>\n<p id=\"fs-id1169739345829\">Thus,<\/p>\n<div id=\"fs-id1169739345832\" class=\"equation unnumbered\">[latex]v(\\frac{\\pi }{4})=-\\frac{1}{\\sqrt{2}}.[\/latex]<\/div>\n<p id=\"fs-id1169739345866\">Next, find [latex]a(t)={v}^{\\prime }(t).[\/latex] Thus, [latex]a(t)={v}^{\\prime }(t)= \\sin t[\/latex] and we have<\/p>\n<div id=\"fs-id1169739345937\" class=\"equation unnumbered\">[latex]a(\\frac{\\pi }{4})=\\frac{1}{\\sqrt{2}}.[\/latex]<\/div>\n<p id=\"fs-id1169739273581\">Since [latex]v(\\frac{\\pi }{4})=-\\frac{1}{\\sqrt{2}}<0[\/latex] and [latex]a(\\frac{\\pi }{4})=\\frac{1}{\\sqrt{2}}>0,[\/latex] we see that velocity and acceleration are acting in opposite directions; that is, the object is being accelerated in the direction opposite to the direction in which it is travelling. Consequently, the particle is slowing down.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739273655\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1169739273659\" class=\"exercise\">\n<div id=\"fs-id1169739273661\" class=\"textbox\">\n<p id=\"fs-id1169739273663\">A block attached to a spring is moving vertically. Its position at time [latex]t[\/latex] is given by [latex]s(t)=2 \\sin t.[\/latex] Find [latex]v(\\frac{5\\pi }{6})[\/latex] and [latex]a(\\frac{5\\pi }{6}).[\/latex] Compare these values and decide whether the block is speeding up or slowing down.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739325513\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739325513\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739325513\">[latex]v(\\frac{5\\pi }{6})=\\text{\u2212}\\sqrt{3}<0[\/latex] and [latex]a(\\frac{5\\pi }{6})=-1<0.[\/latex] The block is speeding up.<\/p>\n<\/div>\n<div id=\"fs-id1169739325579\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1169739325585\">Use <a class=\"autogenerated-content\" href=\"#fs-id1169739376125\">(Figure)<\/a> as a guide.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739325596\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1169739325602\">\n<li>We can find the derivatives of sin [latex]x[\/latex] and cos [latex]x[\/latex] by using the definition of derivative and the limit formulas found earlier. The results are\n<div id=\"fs-id1169736589261\" class=\"equation unnumbered\">[latex]\\frac{d}{dx} \\sin x= \\cos x\\frac{d}{dx} \\cos x=\\text{\u2212} \\sin x.[\/latex]<\/div>\n<\/li>\n<li>With these two formulas, we can determine the derivatives of all six basic trigonometric functions.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169736603422\" class=\"key-equations\">\n<h1>Key Equations<\/h1>\n<ul id=\"fs-id1169736603429\">\n<li><strong>Derivative of sine function<\/strong><br \/>\n[latex]\\frac{d}{dx}( \\sin x)= \\cos x[\/latex]<\/li>\n<li><strong>Derivative of cosine function<\/strong><br \/>\n[latex]\\frac{d}{dx}( \\cos x)=\\text{\u2212} \\sin x[\/latex]<\/li>\n<li><strong>Derivative of tangent function<\/strong><br \/>\n[latex]\\frac{d}{dx}( \\tan x)={ \\sec }^{2}x[\/latex]<\/li>\n<li><strong>Derivative of cotangent function<\/strong><br \/>\n[latex]\\frac{d}{dx}( \\cot x)=\\text{\u2212}{ \\csc }^{2}x[\/latex]<\/li>\n<li><strong>Derivative of secant function<\/strong><br \/>\n[latex]\\frac{d}{dx}( \\sec x)= \\sec x \\tan x[\/latex]<\/li>\n<li><strong>Derivative of cosecant function<\/strong><br \/>\n[latex]\\frac{d}{dx}( \\csc x)=\\text{\u2212} \\csc x \\cot x[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169736597627\" class=\"textbox exercises\">\n<p id=\"fs-id1169736597631\">For the following exercises, find [latex]\\frac{dy}{dx}[\/latex] for the given functions.<\/p>\n<div id=\"fs-id1169736597649\" class=\"exercise\">\n<div id=\"fs-id1169736597652\" class=\"textbox\">\n<p id=\"fs-id1169736597654\">[latex]y={x}^{2}- \\sec x+1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1169736597682\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736597682\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736597682\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{dy}{dx}=2x- \\sec x \\tan x[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736597729\" class=\"exercise\">\n<div id=\"fs-id1169736597731\" class=\"textbox\">\n<p id=\"fs-id1169736597733\">[latex]y=3 \\csc x+\\frac{5}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736658534\" class=\"exercise\">\n<div id=\"fs-id1169736658536\" class=\"textbox\">\n<p id=\"fs-id1169736658538\">[latex]y={x}^{2} \\cot x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736658563\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736658563\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736658563\">[latex]\\frac{dy}{dx}=2x \\cot x-{x}^{2}{ \\csc }^{2}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736658614\" class=\"exercise\">\n<div id=\"fs-id1169736658616\" class=\"textbox\">\n<p id=\"fs-id1169736658618\">[latex]y=x-{x}^{3} \\sin x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739275164\" class=\"exercise\">\n<div id=\"fs-id1169739275166\" class=\"textbox\">\n<p id=\"fs-id1169739275168\">[latex]y=\\frac{ \\sec x}{x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739275191\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739275191\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739275191\">[latex]\\frac{dy}{dx}=\\frac{x \\sec x \\tan x- \\sec x}{{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739275253\" class=\"exercise\">\n<div id=\"fs-id1169739275255\" class=\"textbox\">\n<p id=\"fs-id1169739275257\">[latex]y= \\sin x \\tan x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739303603\" class=\"exercise\">\n<div id=\"fs-id1169739303605\" class=\"textbox\">\n<p id=\"fs-id1169739303607\">[latex]y=(x+ \\cos x)(1- \\sin x)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739303655\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739303655\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739303655\">[latex]\\frac{dy}{dx}=(1- \\sin x)(1- \\sin x)- \\cos x(x+ \\cos x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739303738\" class=\"exercise\">\n<div id=\"fs-id1169739303741\" class=\"textbox\">\n<p id=\"fs-id1169739303743\">[latex]y=\\frac{ \\tan x}{1- \\sec x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739350812\" class=\"exercise\">\n<div id=\"fs-id1169739350814\" class=\"textbox\">\n<p id=\"fs-id1169739350817\">[latex]y=\\frac{1- \\cot x}{1+ \\cot x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739350855\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739350855\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739350855\">[latex]\\frac{dy}{dx}=\\frac{2{ \\csc }^{2}x}{{(1+ \\cot x)}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736593529\" class=\"exercise\">\n<div id=\"fs-id1169736593531\" class=\"textbox\">\n<p id=\"fs-id1169736593533\">[latex]y= \\cos x(1+ \\csc x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169736593662\">For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of [latex]x.[\/latex] Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.<\/p>\n<div id=\"fs-id1169736593674\" class=\"exercise\">\n<div id=\"fs-id1169739266595\" class=\"textbox\">\n<p id=\"fs-id1169739266597\"><strong>[T]<\/strong>[latex]f(x)=\\text{\u2212} \\sin x,x=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739266640\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739266640\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739266640\">[latex]y=\\text{\u2212}x[\/latex]<\/p>\n<p><span id=\"fs-id1169739266655\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205425\/CNX_Calc_Figure_03_05_201.jpg\" alt=\"The graph shows negative sin(x) and the straight line T(x) with slope \u22121 and y intercept 0.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739266669\" class=\"exercise\">\n<div id=\"fs-id1169739266671\" class=\"textbox\">\n<p id=\"fs-id1169739266673\"><strong>[T]<\/strong>[latex]f(x)= \\csc x,x=\\frac{\\pi }{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739266742\" class=\"exercise\">\n<div id=\"fs-id1169739266744\" class=\"textbox\">\n<p id=\"fs-id1169739266746\"><strong>[T]<\/strong>[latex]f(x)=1+ \\cos x,x=\\frac{3\\pi }{2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736655158\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736655158\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736655158\">[latex]y=x+\\frac{2-3\\pi }{2}[\/latex]<\/p>\n<p><span id=\"fs-id1169736655187\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205428\/CNX_Calc_Figure_03_05_203.jpg\" alt=\"The graph shows the cosine function shifted up one and has the straight line T(x) with slope 1 and y intercept (2 \u2013 3\u03c0)\/2.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736655200\" class=\"exercise\">\n<div id=\"fs-id1169736655202\" class=\"textbox\">\n<p id=\"fs-id1169736655204\"><strong>[T]<\/strong>[latex]f(x)= \\sec x,x=\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736655301\" class=\"exercise\">\n<div id=\"fs-id1169736655303\" class=\"textbox\">\n<p id=\"fs-id1169736655305\"><strong>[T]<\/strong>[latex]f(x)={x}^{2}- \\tan xx=0[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>[latex]y=\\text{\u2212}x[\/latex]<\/p>\n<p><span id=\"fs-id1169739305478\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205430\/CNX_Calc_Figure_03_05_205.jpg\" alt=\"The graph shows the function as starting at (\u22121, 3), decreasing to the origin, continuing to slowly decrease to about (1, \u22120.5), at which point it decreases very quickly.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739305492\" class=\"exercise\">\n<div id=\"fs-id1169739305494\" class=\"textbox\">\n<p id=\"fs-id1169739305496\"><strong>[T]<\/strong>[latex]f(x)=5 \\cot xx=\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169739305590\">For the following exercises, find [latex]\\frac{{d}^{2}y}{d{x}^{2}}[\/latex] for the given functions.<\/p>\n<div id=\"fs-id1169739305614\" class=\"exercise\">\n<div id=\"fs-id1169739305617\" class=\"textbox\">\n<p id=\"fs-id1169739305619\">[latex]y=x \\sin x- \\cos x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736662274\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736662274\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736662274\">[latex]3 \\cos x-x \\sin x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736662308\" class=\"exercise\">\n<div id=\"fs-id1169736662310\" class=\"textbox\">\n<p>[latex]y= \\sin x \\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736662371\" class=\"exercise\">\n<div id=\"fs-id1169736662373\" class=\"textbox\">\n<p id=\"fs-id1169736662375\">[latex]y=x-\\frac{1}{2} \\sin x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736662404\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736662404\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736662404\">[latex]\\frac{1}{2} \\sin x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736662423\" class=\"exercise\">\n<div id=\"fs-id1169736662425\" class=\"textbox\">\n<p id=\"fs-id1169736662427\">[latex]y=\\frac{1}{x}+ \\tan x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739303429\" class=\"exercise\">\n<div id=\"fs-id1169739303431\" class=\"textbox\">\n<p id=\"fs-id1169739303433\">[latex]y=2 \\csc x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739303458\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739303458\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739303458\">[latex]\\csc (x)(3{ \\csc }^{2}(x)-1+{ \\cot }^{2}(x))[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739303516\" class=\"exercise\">\n<div id=\"fs-id1169739303518\" class=\"textbox\">\n<p id=\"fs-id1169739303520\">[latex]y={ \\sec }^{2}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739335809\" class=\"exercise\">\n<div id=\"fs-id1169739335812\" class=\"textbox\">\n<p id=\"fs-id1169739335814\">Find all [latex]x[\/latex] values on the graph of [latex]f(x)=-3 \\sin x \\cos x[\/latex] where the tangent line is horizontal.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739335862\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739335862\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739335862\">[latex]\\frac{(2n+1)\\pi }{4},\\text{ where }n\\text{is an integer}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739335906\" class=\"exercise\">\n<div id=\"fs-id1169739335909\" class=\"textbox\">\n<p id=\"fs-id1169739335911\">Find all [latex]x[\/latex] values on the graph of [latex]f(x)=x-2 \\cos x[\/latex] for [latex]0<x<2\\pi[\/latex] where the tangent line has slope 2.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739335992\" class=\"exercise\">\n<div id=\"fs-id1169739335994\" class=\"textbox\">\n<p id=\"fs-id1169739335996\">Let [latex]f(x)= \\cot x.[\/latex] Determine the points on the graph of [latex]f[\/latex] for [latex]0<x<2\\pi[\/latex] where the tangent line(s) is (are) parallel to the line [latex]y=-2x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736613904\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736613904\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736613904\">[latex](\\frac{\\pi }{4},1),(\\frac{3\\pi }{4},-1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736613947\" class=\"exercise\">\n<div id=\"fs-id1169736613949\" class=\"textbox\">\n<p><strong>[T]<\/strong> A mass on a spring bounces up and down in simple harmonic motion, modeled by the function [latex]s(t)=-6 \\cos t[\/latex] where [latex]s[\/latex] is measured in inches and [latex]t[\/latex] is measured in seconds. Find the rate at which the spring is oscillating at [latex]t=5[\/latex] s.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736614037\" class=\"exercise\">\n<div id=\"fs-id1169736614039\" class=\"textbox\">\n<p>Let the position of a swinging pendulum in simple harmonic motion be given by [latex]s(t)=a \\cos t+b \\sin t.[\/latex] Find the constants [latex]a[\/latex] and [latex]b[\/latex] such that when the velocity is 3 cm\/s, [latex]s=0[\/latex] and [latex]t=0.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739341388\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739341388\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739341388\">[latex]a=0,b=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739341409\" class=\"exercise\">\n<div id=\"fs-id1169739341411\" class=\"textbox\">\n<p id=\"fs-id1169739341414\">After a diver jumps off a diving board, the edge of the board oscillates with position given by [latex]s(t)=-5 \\cos t[\/latex] cm at [latex]t[\/latex] seconds after the jump.<\/p>\n<ol id=\"fs-id1169739341452\" style=\"list-style-type: lower-alpha\">\n<li>Sketch one period of the position function for [latex]t\\ge 0.[\/latex]<\/li>\n<li>Find the velocity function.<\/li>\n<li>Sketch one period of the velocity function for [latex]t\\ge 0.[\/latex]<\/li>\n<li>Determine the times when the velocity is 0 over one period.<\/li>\n<li>Find the acceleration function.<\/li>\n<li>Sketch one period of the acceleration function for [latex]t\\ge 0.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739353376\" class=\"exercise\">\n<div id=\"fs-id1169739353378\" class=\"textbox\">\n<p id=\"fs-id1169739353380\">The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by [latex]y=10+5 \\sin x[\/latex] where [latex]y[\/latex] is the number of hamburgers sold and [latex]x[\/latex] represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find [latex]y\\prime[\/latex] and determine the intervals where the number of burgers being sold is increasing.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739353429\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739353429\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739353429\">[latex]{y}^{\\prime }=5 \\cos (x),[\/latex] increasing on [latex](0,\\frac{\\pi }{2}),(\\frac{3\\pi }{2},\\frac{5\\pi }{2}),[\/latex] and [latex](\\frac{7\\pi }{2},12)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739307888\" class=\"exercise\">\n<div id=\"fs-id1169739307890\" class=\"textbox\">\n<p id=\"fs-id1169739307892\"><strong>[T]<\/strong> The amount of rainfall per month in Phoenix, Arizona, can be approximated by [latex]y(t)=0.5+0.3 \\cos t,[\/latex] where [latex]t[\/latex] is months since January. Find [latex]{y}^{\\prime }[\/latex] and use a calculator to determine the intervals where the amount of rain falling is decreasing.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169739264125\">For the following exercises, use the quotient rule to derive the given equations.<\/p>\n<div id=\"fs-id1169739264128\" class=\"exercise\">\n<div id=\"fs-id1169739264130\" class=\"textbox\">\n<p id=\"fs-id1169739264132\">[latex]\\frac{d}{dx}( \\cot x)=\\text{\u2212}{ \\csc }^{2}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739264176\" class=\"exercise\">\n<div id=\"fs-id1169739264178\" class=\"textbox\">\n<p id=\"fs-id1169739264180\">[latex]\\frac{d}{dx}( \\sec x)= \\sec x \\tan x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739264230\" class=\"exercise\">\n<div id=\"fs-id1169739264233\" class=\"textbox\">\n<p id=\"fs-id1169739264235\">[latex]\\frac{d}{dx}( \\csc x)=\\text{\u2212} \\csc x \\cot x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739264287\" class=\"exercise\">\n<div id=\"fs-id1169739264289\" class=\"textbox\">\n<p id=\"fs-id1169739264292\">Use the definition of derivative and the identity<\/p>\n<p id=\"fs-id1169739264295\">[latex]\\cos (x+h)= \\cos x \\cos h- \\sin x \\sin h[\/latex] to prove that [latex]\\frac{d( \\cos x)}{dx}=\\text{\u2212} \\sin x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169739289336\">For the following exercises, find the requested higher-order derivative for the given functions.<\/p>\n<div id=\"fs-id1169739289340\" class=\"exercise\">\n<div id=\"fs-id1169739289343\" class=\"textbox\">\n<p id=\"fs-id1169739289345\">[latex]\\frac{{d}^{3}y}{d{x}^{3}}[\/latex] of [latex]y=3 \\cos x[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p id=\"fs-id1169739289390\">[latex]3 \\sin x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739289409\" class=\"exercise\">\n<div id=\"fs-id1169739289411\" class=\"textbox\">\n<p id=\"fs-id1169739289414\">[latex]\\frac{{d}^{2}y}{d{x}^{2}}[\/latex] of [latex]y=3 \\sin x+{x}^{2} \\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1169736592469\" class=\"textbox\">\n<p id=\"fs-id1169736592471\">[latex]\\frac{{d}^{4}y}{d{x}^{4}}[\/latex] of [latex]y=5 \\cos x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736592517\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736592517\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736592517\">[latex]5 \\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736592536\" class=\"exercise\">\n<div id=\"fs-id1169736592538\" class=\"textbox\">\n<p id=\"fs-id1169736592540\">[latex]\\frac{{d}^{2}y}{d{x}^{2}}[\/latex] of [latex]y= \\sec x+ \\cot x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739301937\" class=\"exercise\">\n<div id=\"fs-id1169739301939\" class=\"textbox\">\n<p id=\"fs-id1169739301941\">[latex]\\frac{{d}^{3}y}{d{x}^{3}}[\/latex] of [latex]y={x}^{10}- \\sec x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739301990\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739301990\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739301990\">[latex]720{x}^{7}-5 \\tan (x){ \\sec }^{3}(x)-{ \\tan }^{3}(x) \\sec (x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":311,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1847","chapter","type-chapter","status-publish","hentry"],"part":1777,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1847","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1847\/revisions"}],"predecessor-version":[{"id":2515,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1847\/revisions\/2515"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/parts\/1777"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1847\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/media?parent=1847"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapter-type?post=1847"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/contributor?post=1847"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/license?post=1847"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}