{"id":348,"date":"2021-02-04T01:17:12","date_gmt":"2021-02-04T01:17:12","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=348"},"modified":"2022-03-16T05:33:45","modified_gmt":"2022-03-16T05:33:45","slug":"higher-order-derivatives-of-trig-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/higher-order-derivatives-of-trig-functions\/","title":{"raw":"Higher-Order Derivatives of Trig Functions","rendered":"Higher-Order Derivatives of Trig Functions"},"content":{"raw":"<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=\"textbook exercises\">\r\n<h3>Example: Finding Higher-Order Derivatives of [latex]y= \\sin x[\/latex]<\/h3>\r\n<p id=\"fs-id1169739303850\">Find the first four derivatives of [latex]y= \\sin x[\/latex].<\/p>\r\n[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}{lll} y &amp; = &amp; \\sin x \\\\ \\frac{dy}{dx} &amp; = &amp; \\cos x \\\\ \\frac{d^2 y}{dx^2} &amp; = &amp; \u2212\\sin x \\\\ \\frac{d^3 y}{dx^3} &amp; = &amp; \u2212\\cos x \\\\ \\frac{d^4 y}{dx^4} &amp; = &amp; \\sin x \\end{array}[\/latex]<\/div>\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 [latex]\\sin x[\/latex] equals [latex]\\sin x[\/latex], so<\/p>\r\n\r\n<div id=\"fs-id1169739284998\" class=\"equation unnumbered\">[latex]\\begin{array}{l}\\frac{d^4}{dx^4}(\\sin x)=\\frac{d^8}{dx^8}(\\sin x)=\\frac{d^{12}}{dx^{12}}(\\sin x)=\\cdots =\\frac{d^{4n}}{dx^{4n}}(\\sin x)= \\sin x \\\\ \\frac{d^5}{dx^5}(\\sin x)=\\frac{d^9}{dx^9}(\\sin x)=\\frac{d^{13}}{dx^{13}}(\\sin x)=\\cdots =\\frac{d^{4n+1}}{dx^{4n+1}}(\\sin x)= \\cos x\\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739298009\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739298017\">For [latex]y= \\cos x[\/latex], find [latex]\\dfrac{d^4 y}{dx^4}[\/latex].<\/p>\r\n[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<h4>Hint<\/h4>\r\n<p id=\"fs-id1169739298080\">See the previous example.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Finding Higher-Order Derivatives of [latex]y= \\sin x[\/latex] and the above Try It.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=1316&amp;end=1403&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesOfTrigonometricFunctions1316to1403_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.5 Derivatives of Trigonometric Functions (edited)\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1169739298086\" class=\"textbook exercises\">\r\n<h3>Example: Using the Pattern for Higher-Order Derivatives of [latex]y= \\sin x[\/latex]<\/h3>\r\n<p id=\"fs-id1169739293614\">Find [latex]\\frac{d^{74}}{dx^{74}}(\\sin x)[\/latex].<\/p>\r\n[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}}{dx^{74}}(\\sin x)=\\frac{d^{72+2}}{dx^{72+2}}(\\sin x)=\\frac{d^2}{dx^2}(\\sin x)=\u2212\\sin x[\/latex].[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736595960\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169736595968\">For [latex]y= \\sin x[\/latex], find [latex]\\frac{d^{59}}{dx^{59}}(\\sin x)[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169736596026\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736596026\"]\r\n<p id=\"fs-id1169736596026\">[latex]\u2212\\cos x[\/latex]<\/p>\r\n\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169736596046\">[latex]\\frac{d^{59}}{dx^{59}}(\\sin x)=\\frac{d^{4(14)+3}}{dx^{4(14)+3}}(\\sin x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]224399[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739376125\" class=\"textbook exercises\">\r\n<h3>Example: An Application to Acceleration<\/h3>\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].<\/p>\r\nFind [latex]v\\left(\\frac{\\pi}{4}\\right)[\/latex]\u00a0 and\u00a0 [latex]a\\left(\\frac{\\pi}{4}\\right)[\/latex]. Compare these values and decide whether the particle is speeding up or slowing down.\r\n<p id=\"fs-id1169739273581\">[reveal-answer q=\"519394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"519394\"]<\/p>\r\nFirst find [latex]v(t)=s^{\\prime}(t)[\/latex]: [latex]v(t)=s^{\\prime}(t)=\u2212\\cos t[\/latex]. Thus, [latex]v\\left(\\frac{\\pi}{4}\\right)=-\\frac{1}{\\sqrt{2}}[\/latex].\r\n\r\nNext, find [latex]a(t)=v^{\\prime}(t)[\/latex].\r\n\r\nThus, [latex]a(t)=v^{\\prime}(t)= \\sin t[\/latex] and we have [latex]a\\left(\\frac{\\pi}{4}\\right)=\\frac{1}{\\sqrt{2}}[\/latex].\r\n\r\nSince [latex]v\\left(\\frac{\\pi}{4}\\right)=-\\frac{1}{\\sqrt{2}}&lt;0[\/latex] and [latex]a\\left(\\frac{\\pi}{4}\\right)=\\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 traveling.\r\n\r\nConsequently, the particle is slowing down.[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: An Application to Acceleration.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=1492&amp;end=1573&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesOfTrigonometricFunctions1492to1573_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.5 Derivatives of Trigonometric Functions (edited)\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1169739273655\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\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\nFind [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.\r\n\r\n[reveal-answer q=\"fs-id1169739325513\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739325513\"]\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\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169739325585\">Use the last example as a guide.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<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=\"textbook exercises\">\n<h3>Example: Finding Higher-Order Derivatives of [latex]y= \\sin x[\/latex]<\/h3>\n<p id=\"fs-id1169739303850\">Find the first four derivatives of [latex]y= \\sin x[\/latex].<\/p>\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}{lll} y & = & \\sin x \\\\ \\frac{dy}{dx} & = & \\cos x \\\\ \\frac{d^2 y}{dx^2} & = & \u2212\\sin x \\\\ \\frac{d^3 y}{dx^3} & = & \u2212\\cos x \\\\ \\frac{d^4 y}{dx^4} & = & \\sin x \\end{array}[\/latex]<\/div>\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 [latex]\\sin x[\/latex] equals [latex]\\sin x[\/latex], so<\/p>\n<div id=\"fs-id1169739284998\" class=\"equation unnumbered\">[latex]\\begin{array}{l}\\frac{d^4}{dx^4}(\\sin x)=\\frac{d^8}{dx^8}(\\sin x)=\\frac{d^{12}}{dx^{12}}(\\sin x)=\\cdots =\\frac{d^{4n}}{dx^{4n}}(\\sin x)= \\sin x \\\\ \\frac{d^5}{dx^5}(\\sin x)=\\frac{d^9}{dx^9}(\\sin x)=\\frac{d^{13}}{dx^{13}}(\\sin x)=\\cdots =\\frac{d^{4n+1}}{dx^{4n+1}}(\\sin x)= \\cos x\\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739298009\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739298017\">For [latex]y= \\cos x[\/latex], find [latex]\\dfrac{d^4 y}{dx^4}[\/latex].<\/p>\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<h4>Hint<\/h4>\n<p id=\"fs-id1169739298080\">See the previous example.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Finding Higher-Order Derivatives of [latex]y= \\sin x[\/latex] and the above Try It.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=1316&amp;end=1403&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesOfTrigonometricFunctions1316to1403_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.5 Derivatives of Trigonometric Functions (edited)&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1169739298086\" class=\"textbook exercises\">\n<h3>Example: Using the Pattern for Higher-Order Derivatives of [latex]y= \\sin x[\/latex]<\/h3>\n<p id=\"fs-id1169739293614\">Find [latex]\\frac{d^{74}}{dx^{74}}(\\sin x)[\/latex].<\/p>\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}}{dx^{74}}(\\sin x)=\\frac{d^{72+2}}{dx^{72+2}}(\\sin x)=\\frac{d^2}{dx^2}(\\sin x)=\u2212\\sin x[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736595960\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169736595968\">For [latex]y= \\sin x[\/latex], find [latex]\\frac{d^{59}}{dx^{59}}(\\sin x)[\/latex].<\/p>\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]\u2212\\cos x[\/latex]<\/p>\n<h4>Hint<\/h4>\n<p id=\"fs-id1169736596046\">[latex]\\frac{d^{59}}{dx^{59}}(\\sin x)=\\frac{d^{4(14)+3}}{dx^{4(14)+3}}(\\sin x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm224399\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=224399&theme=oea&iframe_resize_id=ohm224399&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1169739376125\" class=\"textbook exercises\">\n<h3>Example: An Application to Acceleration<\/h3>\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].<\/p>\n<p>Find [latex]v\\left(\\frac{\\pi}{4}\\right)[\/latex]\u00a0 and\u00a0 [latex]a\\left(\\frac{\\pi}{4}\\right)[\/latex]. Compare these values and decide whether the particle is speeding up or slowing down.<\/p>\n<p id=\"fs-id1169739273581\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q519394\">Show Solution<\/span><\/p>\n<div id=\"q519394\" class=\"hidden-answer\" style=\"display: none\">\n<p>First find [latex]v(t)=s^{\\prime}(t)[\/latex]: [latex]v(t)=s^{\\prime}(t)=\u2212\\cos t[\/latex]. Thus, [latex]v\\left(\\frac{\\pi}{4}\\right)=-\\frac{1}{\\sqrt{2}}[\/latex].<\/p>\n<p>Next, find [latex]a(t)=v^{\\prime}(t)[\/latex].<\/p>\n<p>Thus, [latex]a(t)=v^{\\prime}(t)= \\sin t[\/latex] and we have [latex]a\\left(\\frac{\\pi}{4}\\right)=\\frac{1}{\\sqrt{2}}[\/latex].<\/p>\n<p>Since [latex]v\\left(\\frac{\\pi}{4}\\right)=-\\frac{1}{\\sqrt{2}}<0[\/latex] and [latex]a\\left(\\frac{\\pi}{4}\\right)=\\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 traveling.<\/p>\n<p>Consequently, the particle is slowing down.<\/p><\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: An Application to Acceleration.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=1492&amp;end=1573&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesOfTrigonometricFunctions1492to1573_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.5 Derivatives of Trigonometric Functions (edited)&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1169739273655\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\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<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\\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<h4>Hint<\/h4>\n<p id=\"fs-id1169739325585\">Use the last example as a guide.<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-348\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>3.5 Derivatives of Trigonometric Functions (edited). <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"3.5 Derivatives of Trigonometric Functions (edited)\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-348","chapter","type-chapter","status-publish","hentry"],"part":35,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/348","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":17,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/348\/revisions"}],"predecessor-version":[{"id":4813,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/348\/revisions\/4813"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/35"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/348\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=348"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=348"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=348"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=348"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}