{"id":3294,"date":"2024-06-18T18:07:01","date_gmt":"2024-06-18T18:07:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3294"},"modified":"2024-08-05T01:56:55","modified_gmt":"2024-08-05T01:56:55","slug":"derivatives-of-trigonometric-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/derivatives-of-trigonometric-functions-fresh-take\/","title":{"raw":"Derivatives of Trigonometric Functions: Fresh Take","rendered":"Derivatives of Trigonometric Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Calculate the derivatives of sine and cosine functions, including second derivatives and beyond<\/li>\r\n\t<li>Determine the derivatives for basic trig functions like tangent, cotangent, secant, and cosecant<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Derivatives of the Sine and Cosine Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Derivative of Sine: [latex]\\frac{d}{dx}(\\sin x) = \\cos x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Derivative of Cosine: [latex]\\frac{d}{dx}(\\cos x) = -\\sin x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Graphical Interpretation:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Where sine has a maximum or minimum, cosine (its derivative) is zero<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Where cosine has a maximum or minimum, negative sine (its derivative) is zero<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169736660715\">Find the derivative of [latex]f(x)= \\sin x \\cos x.[\/latex]<\/p>\r\n<p>[reveal-answer q=\"300277\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"300277\"]<\/p>\r\n<p id=\"fs-id1169738821957\">Don\u2019t forget to use the product rule.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739042083\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739042083\"]<\/p>\r\n<p id=\"fs-id1169739042083\">[latex]f^{\\prime}(x)=\\cos^2 x-\\sin^2 x[\/latex]<\/p>\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=183&amp;end=247&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesOfTrigonometricFunctions183to247_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.5 Derivatives of Trigonometric Functions (edited)\" here (opens in new window)<\/a>.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738971415\">Find the derivative of [latex]f(x)=\\dfrac{x}{\\cos x}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"488399\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"488399\"]<\/p>\r\n<p id=\"fs-id1169736587923\">Use the quotient rule.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739286412\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739286412\"]<\/p>\r\n<p id=\"fs-id1169739286412\">[latex]\\dfrac{\\cos x+x \\sin x}{\\cos^2 x}[\/latex]<\/p>\r\n<p>Watch the following video to see the worked solution to this example<\/p>\r\n<p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=699&amp;end=766&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesOfTrigonometricFunctions699to766_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.5 Derivatives of Trigonometric Functions (edited)\" here (opens in new window)<\/a>.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]f(x) = 5x^3 \\sin x[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"315976\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"315976\"]\u00a0<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Using the product rule:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] \\begin{array}{rcl} f'(x) &amp;=&amp; \\frac{d}{dx}(5x^3) \\cdot \\sin x + 5x^3 \\cdot \\frac{d}{dx}(\\sin x) \\\\ &amp;=&amp; 15x^2 \\cdot \\sin x + 5x^3 \\cdot \\cos x \\\\ &amp;=&amp; 15x^2 \\sin x + 5x^3 \\cos x \\end{array} [\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>A particle moves along a coordinate axis with position [latex]s(t) = 2\\sin t - t[\/latex] for [latex]0 \\leq t \\leq 2\\pi[\/latex]. At what times is the particle at rest?<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"585046\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"585046\"]<\/p>\r\n<p>Find velocity function:<\/p>\r\n<p style=\"text-align: center;\">[latex]v(t) = s'(t) = 2\\cos t - 1[\/latex]<\/p>\r\n<p>Set velocity to zero:<\/p>\r\n<p style=\"text-align: center;\">[latex]2\\cos t - 1 = 0[\/latex]<\/p>\r\n<p>Solve:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\cos t = \\frac{1}{2}[\/latex]<\/p>\r\n<p>In the given interval:<\/p>\r\n<p style=\"text-align: center;\">[latex]t = \\frac{\\pi}{3}[\/latex] and [latex]t = \\frac{5\\pi}{3}[\/latex]<\/p>\r\n<p>The particle is at rest at [latex]t = \\frac{\\pi}{3}[\/latex] and [latex]t = \\frac{5\\pi}{3}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\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<p>[reveal-answer q=\"fs-id1169739274425\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739274425\"]<\/p>\r\n<p id=\"fs-id1169739274425\">[latex]t=\\frac{\\pi}{3}, \\, t=\\frac{2\\pi}{3}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Derivatives of Other Trigonometric Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Derivatives of Other Trigonometric Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(\\tan x) = \\sec^2 x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(\\cot x) = -\\csc^2 x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(\\sec x) = \\sec x \\tan x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(\\csc x) = -\\csc x \\cot x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Derivation Methods:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Use quotient rule for [latex]\\tan[\/latex] and [latex]\\cot[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply chain rule and trigonometric identities for [latex]\\sec[\/latex] and [latex]\\csc[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739188150\">Find the derivative of [latex]f(x)=2 \\tan x-3 \\cot x[\/latex].<\/p>\r\n<p>[reveal-answer q=\"661109\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"661109\"]<\/p>\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<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739188198\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739188198\"]<\/p>\r\n<p id=\"fs-id1169739188198\">[latex]f^{\\prime}(x)=2 \\sec^2 x+3 \\csc^2 x[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Find the derivative of [latex]f(x) = \\csc x + x \\tan x[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"739702\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"739702\"]<\/p>\r\n<p>Use the sum rule:<\/p>\r\n<p style=\"text-align: center;\">[latex]f'(x) = \\frac{d}{dx}(\\csc x) + \\frac{d}{dx}(x \\tan x)[\/latex]<\/p>\r\n<p>Apply known derivative and product rule:<\/p>\r\n<p style=\"text-align: center;\">[latex]f'(x) = -\\csc x \\cot x + (1 \\cdot \\tan x + x \\cdot \\sec^2 x)[\/latex]<\/p>\r\n<p>Simplify:<\/p>\r\n<p style=\"text-align: center;\">[latex]f'(x) = -\\csc x \\cot x + \\tan x + x \\sec^2 x[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\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>\r\n<p>[reveal-answer q=\"6078833\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"6078833\"]<\/p>\r\n<p id=\"fs-id1169739303758\">Evaluate the derivative at [latex]x=\\frac{\\pi}{6}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169736662675\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169736662675\"]<\/p>\r\n<p id=\"fs-id1169736662675\">[latex]\\frac{4}{3}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Find the equation of a line tangent to the graph of [latex]f(x) = \\cot x[\/latex] at [latex]x = \\frac{\\pi}{4}[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"310861\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"310861\"]<\/p>\r\n<p>Find the point: [latex]f(\\frac{\\pi}{4}) = \\cot \\frac{\\pi}{4} = 1[\/latex]<\/p>\r\n<p style=\"text-align: center;\">Point: [latex](\\frac{\\pi}{4}, 1)[\/latex]<\/p>\r\n<p>Find the slope:<\/p>\r\n<p style=\"text-align: center;\">[latex]f'(x) = -\\csc^2 x[\/latex]<br \/>\r\n[latex]f'(\\frac{\\pi}{4}) = -\\csc^2 \\frac{\\pi}{4} = -2[\/latex]<\/p>\r\n<p>Use point-slope form:<\/p>\r\n<p style=\"text-align: center;\">[latex]y - 1 = -2(x - \\frac{\\pi}{4})[\/latex]<\/p>\r\n<p>Simplify to slope-intercept form:<\/p>\r\n<p style=\"text-align: center;\">[latex]y = -2x + 1 + \\frac{\\pi}{2}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Higher-Order Derivatives of Trig Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Cyclic Pattern of Derivatives:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For [latex]\\sin x[\/latex]: [latex]\\sin x \\to \\cos x \\to -\\sin x \\to -\\cos x \\to \\sin x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For [latex]\\cos x[\/latex]: [latex]\\cos x \\to -\\sin x \\to -\\cos x \\to \\sin x \\to \\cos x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Predicting Higher-Order Derivatives:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Use the remainder when the derivative order is divided by [latex]4[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pattern repeats every four derivatives<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p><strong>Problem-Solving Strategy<\/strong><\/p>\r\n<ol>\r\n\t<li class=\"whitespace-normal break-words\">Find the order of the derivative ([latex]n[\/latex])<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Calculate [latex]n \\bmod 4[\/latex] (remainder when [latex]n[\/latex] is divided by [latex]4[\/latex])<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use the remainder to determine the derivative:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For [latex]\\sin x[\/latex]:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Remainder [latex]0[\/latex]: [latex]\\sin x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remainder [latex]1[\/latex]: [latex]\\cos x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remainder [latex]2[\/latex]: [latex]-\\sin x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remainder [latex]3[\/latex]: [latex]-\\cos x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For [latex]\\cos x[\/latex]:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Remainder [latex]0[\/latex]: [latex]\\cos x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remainder [latex]1[\/latex]: [latex]-\\sin x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remainder [latex]2[\/latex]: [latex]-\\cos x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remainder [latex]3[\/latex]: [latex]\\sin x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169736595968\">For [latex]y= \\sin x[\/latex], find [latex]\\dfrac{d^{59}}{dx^{59}}(\\sin x)[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"231540\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"231540\"][latex]\\dfrac{d^{59}}{dx^{59}}(\\sin x)=\\dfrac{d^{4(14)+3}}{dx^{4(14)+3}}(\\sin x)[\/latex][\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169736596026\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169736596026\"]<\/p>\r\n<p id=\"fs-id1169736596026\">[latex]\u2212\\cos x[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Find [latex]\\dfrac{d^{74}}{dx^{74}}(\\sin x)[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"232154\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"232154\"]<\/p>\r\n<p>[latex]74 \\div 4 = 18[\/latex] remainder [latex]2[\/latex]<\/p>\r\n<p>For [latex]\\sin x[\/latex], remainder [latex]2[\/latex] corresponds to [latex]-\\sin x[\/latex]<\/p>\r\n<p>Therefore, [latex]\\dfrac{d^{74}}{dx^{74}}(\\sin x) = -\\sin x[\/latex][\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\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].<\/p>\r\n<p>Find [latex]v\\left(\\frac{5\\pi}{6}\\right)[\/latex] and [latex]a\\left(\\frac{5\\pi}{6}\\right)[\/latex]. Compare these values and decide whether the block is speeding up or slowing down.<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739325513\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739325513\"]<\/p>\r\n<p id=\"fs-id1169739325513\">[latex]v\\left(\\frac{5\\pi}{6}\\right)=\u2212\\sqrt{3}&lt;0[\/latex]\u00a0 and\u00a0 [latex]a\\left(\\frac{5\\pi}{6}\\right)=-1&lt;0[\/latex]. The block is speeding up.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate the derivatives of sine and cosine functions, including second derivatives and beyond<\/li>\n<li>Determine the derivatives for basic trig functions like tangent, cotangent, secant, and cosecant<\/li>\n<\/ul>\n<\/section>\n<h2>Derivatives of the Sine and Cosine Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Derivative of Sine: [latex]\\frac{d}{dx}(\\sin x) = \\cos x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Derivative of Cosine: [latex]\\frac{d}{dx}(\\cos x) = -\\sin x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Graphical Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Where sine has a maximum or minimum, cosine (its derivative) is zero<\/li>\n<li class=\"whitespace-normal break-words\">Where cosine has a maximum or minimum, negative sine (its derivative) is zero<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169736660715\">Find the derivative of [latex]f(x)= \\sin x \\cos x.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q300277\">Hint<\/button><\/p>\n<div id=\"q300277\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738821957\">Don\u2019t forget to use the product rule.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739042083\">Show Solution<\/button><\/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<p>Watch the following video to see the worked solution to this example.<\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=183&amp;end=247&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesOfTrigonometricFunctions183to247_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.5 Derivatives of Trigonometric Functions (edited)&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738971415\">Find the derivative of [latex]f(x)=\\dfrac{x}{\\cos x}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q488399\">Hint<\/button><\/p>\n<div id=\"q488399\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736587923\">Use the quotient rule.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739286412\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739286412\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739286412\">[latex]\\dfrac{\\cos x+x \\sin x}{\\cos^2 x}[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example<\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=699&amp;end=766&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesOfTrigonometricFunctions699to766_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.5 Derivatives of Trigonometric Functions (edited)&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]f(x) = 5x^3 \\sin x[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q315976\">Show Answer<\/button><\/p>\n<div id=\"q315976\" class=\"hidden-answer\" style=\"display: none\">\u00a0<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Using the product rule:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{array}{rcl} f'(x) &=& \\frac{d}{dx}(5x^3) \\cdot \\sin x + 5x^3 \\cdot \\frac{d}{dx}(\\sin x) \\\\ &=& 15x^2 \\cdot \\sin x + 5x^3 \\cdot \\cos x \\\\ &=& 15x^2 \\sin x + 5x^3 \\cos x \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>A particle moves along a coordinate axis with position [latex]s(t) = 2\\sin t - t[\/latex] for [latex]0 \\leq t \\leq 2\\pi[\/latex]. At what times is the particle at rest?<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q585046\">Show Answer<\/button><\/p>\n<div id=\"q585046\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find velocity function:<\/p>\n<p style=\"text-align: center;\">[latex]v(t) = s'(t) = 2\\cos t - 1[\/latex]<\/p>\n<p>Set velocity to zero:<\/p>\n<p style=\"text-align: center;\">[latex]2\\cos t - 1 = 0[\/latex]<\/p>\n<p>Solve:<\/p>\n<p style=\"text-align: center;\">[latex]\\cos t = \\frac{1}{2}[\/latex]<\/p>\n<p>In the given interval:<\/p>\n<p style=\"text-align: center;\">[latex]t = \\frac{\\pi}{3}[\/latex] and [latex]t = \\frac{5\\pi}{3}[\/latex]<\/p>\n<p>The particle is at rest at [latex]t = \\frac{\\pi}{3}[\/latex] and [latex]t = \\frac{5\\pi}{3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739274425\">Show Solution<\/button><\/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>\n<\/section>\n<h2>Derivatives of Other Trigonometric Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Derivatives of Other Trigonometric Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(\\tan x) = \\sec^2 x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(\\cot x) = -\\csc^2 x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(\\sec x) = \\sec x \\tan x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(\\csc x) = -\\csc x \\cot x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Derivation Methods:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use quotient rule for [latex]\\tan[\/latex] and [latex]\\cot[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Apply chain rule and trigonometric identities for [latex]\\sec[\/latex] and [latex]\\csc[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739188150\">Find the derivative of [latex]f(x)=2 \\tan x-3 \\cot x[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q661109\">Hint<\/button><\/p>\n<div id=\"q661109\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736662619\">Use the rule for differentiating a constant multiple and the rule for differentiating a difference of two functions.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739188198\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739188198\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739188198\">[latex]f^{\\prime}(x)=2 \\sec^2 x+3 \\csc^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find the derivative of [latex]f(x) = \\csc x + x \\tan x[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q739702\">Show Answer<\/button><\/p>\n<div id=\"q739702\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the sum rule:<\/p>\n<p style=\"text-align: center;\">[latex]f'(x) = \\frac{d}{dx}(\\csc x) + \\frac{d}{dx}(x \\tan x)[\/latex]<\/p>\n<p>Apply known derivative and product rule:<\/p>\n<p style=\"text-align: center;\">[latex]f'(x) = -\\csc x \\cot x + (1 \\cdot \\tan x + x \\cdot \\sec^2 x)[\/latex]<\/p>\n<p>Simplify:<\/p>\n<p style=\"text-align: center;\">[latex]f'(x) = -\\csc x \\cot x + \\tan x + x \\sec^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q6078833\">Hint<\/button><\/p>\n<div id=\"q6078833\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739303758\">Evaluate the derivative at [latex]x=\\frac{\\pi}{6}[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169736662675\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169736662675\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736662675\">[latex]\\frac{4}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find the equation of a line tangent to the graph of [latex]f(x) = \\cot x[\/latex] at [latex]x = \\frac{\\pi}{4}[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q310861\">Show Answer<\/button><\/p>\n<div id=\"q310861\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the point: [latex]f(\\frac{\\pi}{4}) = \\cot \\frac{\\pi}{4} = 1[\/latex]<\/p>\n<p style=\"text-align: center;\">Point: [latex](\\frac{\\pi}{4}, 1)[\/latex]<\/p>\n<p>Find the slope:<\/p>\n<p style=\"text-align: center;\">[latex]f'(x) = -\\csc^2 x[\/latex]<br \/>\n[latex]f'(\\frac{\\pi}{4}) = -\\csc^2 \\frac{\\pi}{4} = -2[\/latex]<\/p>\n<p>Use point-slope form:<\/p>\n<p style=\"text-align: center;\">[latex]y - 1 = -2(x - \\frac{\\pi}{4})[\/latex]<\/p>\n<p>Simplify to slope-intercept form:<\/p>\n<p style=\"text-align: center;\">[latex]y = -2x + 1 + \\frac{\\pi}{2}[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/section>\n<h2>Higher-Order Derivatives of Trig Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Cyclic Pattern of Derivatives:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]\\sin x[\/latex]: [latex]\\sin x \\to \\cos x \\to -\\sin x \\to -\\cos x \\to \\sin x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For [latex]\\cos x[\/latex]: [latex]\\cos x \\to -\\sin x \\to -\\cos x \\to \\sin x \\to \\cos x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Predicting Higher-Order Derivatives:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use the remainder when the derivative order is divided by [latex]4[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Pattern repeats every four derivatives<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol>\n<li class=\"whitespace-normal break-words\">Find the order of the derivative ([latex]n[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]n \\bmod 4[\/latex] (remainder when [latex]n[\/latex] is divided by [latex]4[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Use the remainder to determine the derivative:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]\\sin x[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Remainder [latex]0[\/latex]: [latex]\\sin x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Remainder [latex]1[\/latex]: [latex]\\cos x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Remainder [latex]2[\/latex]: [latex]-\\sin x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Remainder [latex]3[\/latex]: [latex]-\\cos x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For [latex]\\cos x[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Remainder [latex]0[\/latex]: [latex]\\cos x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Remainder [latex]1[\/latex]: [latex]-\\sin x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Remainder [latex]2[\/latex]: [latex]-\\cos x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Remainder [latex]3[\/latex]: [latex]\\sin x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169736595968\">For [latex]y= \\sin x[\/latex], find [latex]\\dfrac{d^{59}}{dx^{59}}(\\sin x)[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q231540\">Hint<\/button><\/p>\n<div id=\"q231540\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{d^{59}}{dx^{59}}(\\sin x)=\\dfrac{d^{4(14)+3}}{dx^{4(14)+3}}(\\sin x)[\/latex]<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169736596026\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169736596026\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736596026\">[latex]\u2212\\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find [latex]\\dfrac{d^{74}}{dx^{74}}(\\sin x)[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q232154\">Show Answer<\/button><\/p>\n<div id=\"q232154\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]74 \\div 4 = 18[\/latex] remainder [latex]2[\/latex]<\/p>\n<p>For [latex]\\sin x[\/latex], remainder [latex]2[\/latex] corresponds to [latex]-\\sin x[\/latex]<\/p>\n<p>Therefore, [latex]\\dfrac{d^{74}}{dx^{74}}(\\sin x) = -\\sin x[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\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].<\/p>\n<p>Find [latex]v\\left(\\frac{5\\pi}{6}\\right)[\/latex] and [latex]a\\left(\\frac{5\\pi}{6}\\right)[\/latex]. Compare these values and decide whether the block is speeding up or slowing down.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739325513\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739325513\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739325513\">[latex]v\\left(\\frac{5\\pi}{6}\\right)=\u2212\\sqrt{3}<0[\/latex]\u00a0 and\u00a0 [latex]a\\left(\\frac{5\\pi}{6}\\right)=-1<0[\/latex]. The block is speeding up.<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":560,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3294"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3294\/revisions"}],"predecessor-version":[{"id":3748,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3294\/revisions\/3748"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/560"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3294\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3294"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3294"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3294"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3294"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}