{"id":3350,"date":"2024-06-20T15:02:22","date_gmt":"2024-06-20T15:02:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3350"},"modified":"2024-08-05T02:01:07","modified_gmt":"2024-08-05T02:01:07","slug":"derivatives-of-inverse-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/derivatives-of-inverse-functions-fresh-take\/","title":{"raw":"Derivatives of Inverse Functions: Fresh Take","rendered":"Derivatives of Inverse Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Find the derivative of an inverse function<\/li>\r\n\t<li>Identify the derivatives for inverse trig functions like arcsine, arccosine, and arctangent<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Derivatives of Various Inverse 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\">Inverse Function Theorem:\r\n\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 invertible and differentiable function f(x): [latex]\\dfrac{1}{f^{\\prime}(f^{-1}(x))}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Graphical Interpretation:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Tangent lines of [latex]f(x)[\/latex] and [latex]f^(-1)(x)[\/latex] have reciprocal slopes<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Symmetric about [latex]y = x[\/latex] line<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Extending the Power Rule:\r\n\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 positive integer [latex]n[\/latex]: [latex]\\frac{d}{dx}(x^{1\/n}) = \\frac{1}{n}x^{(1\/n)-1}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For positive integer [latex]n[\/latex] and any integer m: [latex]\\frac{d}{dx}(x^{m\/n}) = \\frac{m}{n}x^{(m\/n)-1}[\/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-id1169739269403\">Use the inverse function theorem to find the derivative of [latex]g(x)=\\dfrac{1}{x+2}.[\/latex] Compare the result obtained by differentiating [latex]g(x)[\/latex] directly.<\/p>\r\n<p>[reveal-answer q=\"336869\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"336869\"]<br \/>\r\n[latex]g^{\\prime}(x)=-\\frac{1}{(x+2)^2}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739036425\">Use the inverse function theorem to find the derivative of [latex]g(x)=\\sqrt[3]{x}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"fs-id1169738969956\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169738969956\"]<\/p>\r\n<p id=\"fs-id1169738969956\">The function [latex]g(x)=\\sqrt[3]{x}[\/latex] is the inverse of the function [latex]f(x)=x^3[\/latex]. Since [latex]g^{\\prime}(x)=\\frac{1}{f^{\\prime}(g(x))}[\/latex], begin by finding [latex]f^{\\prime}(x)[\/latex]. Thus,<\/p>\r\n<div id=\"fs-id1169739305067\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=3x^2[\/latex] and [latex]f^{\\prime}(g(x))=3(\\sqrt[3]{x})^2=3x^{2\/3}[\/latex]<\/div>\r\n<p id=\"fs-id1169739034112\">Finally,<\/p>\r\n<div id=\"fs-id1169739340252\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]g^{\\prime}(x)=\\frac{1}{3x^{2\/3}}=\\frac{1}{3}x^{-2\/3}[\/latex]<\/div>\r\n\r\n[\/hidden-answer]\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169736607624\">Find the derivative of [latex]s(t)=\\sqrt{2t+1}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"8665521\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"8665521\"]<\/p>\r\n<p id=\"fs-id1169736596062\">Rewrite as [latex]s(t)=(2t+1)^{1\/2}[\/latex] and use the chain rule.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739183503\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739183503\"]<\/p>\r\n<p id=\"fs-id1169739183503\">[latex]s^{\\prime}(t)=(2t+1)^{\u22121\/2}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Derivatives of Inverse 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\">Key Derivatives:<br \/>\r\n<center>[latex]\\begin{array}{lllll}\\frac{d}{dx}(\\sin^{-1} x)=\\large \\frac{1}{\\sqrt{1-x^2}} &amp; &amp; &amp; &amp; \\frac{d}{dx}(\\cos^{-1} x)=\\large \\frac{-1}{\\sqrt{1-x^2}} \\\\ \\frac{d}{dx}(\\tan^{-1} x)=\\large \\frac{1}{1+x^2} &amp; &amp; &amp; &amp; \\frac{d}{dx}(\\cot^{-1} x)=\\large \\frac{-1}{1+x^2} \\\\ \\frac{d}{dx}(\\sec^{-1} x)=\\large \\frac{1}{|x|\\sqrt{x^2-1}} &amp; &amp; &amp; &amp; \\frac{d}{dx}(\\csc^{-1} x)=\\large \\frac{-1}{|x|\\sqrt{x^2-1}} \\end{array}[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">For composite functions like [latex]\\sin^{-1}(g(x))[\/latex], use the chain rule in conjunction with inverse trig derivatives<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to the domains of inverse trigonometric functions when differentiating<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739111210\">Use the inverse function theorem to find the derivative of [latex]g(x)=\\tan^{-1} x[\/latex].<\/p>\r\n<p>[reveal-answer q=\"462877\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"462877\"]<\/p>\r\n<p id=\"fs-id1169736613675\">The inverse of [latex]g(x)[\/latex] is [latex]f(x)= \\tan x[\/latex].\u00a0<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169736659262\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169736659262\"]<\/p>\r\n<p id=\"fs-id1169736659262\">[latex]g^{\\prime}(x)=\\dfrac{1}{1+x^2}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739027843\">Find the derivative of [latex]h(x)=x^2 \\sin^{-1} x[\/latex]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169736603526\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169736603526\"]<\/p>\r\n<p id=\"fs-id1169736603526\">By applying the product rule, we have<\/p>\r\n<div id=\"fs-id1169736603529\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=2x \\sin^{-1} x+\\frac{1}{\\sqrt{1-x^2}} \\cdot x^2[\/latex].<\/div>\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0dlA5QJZYsw?controls=0&amp;start=786&amp;end=848&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\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\/3.7DerivativesOfInverseFuunctions786to848_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.7 Derivatives of Inverse 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-id1169739325671\">Find the derivative of [latex]h(x)= \\cos^{-1} (3x-1)[\/latex]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739304017\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739304017\"]<\/p>\r\n<p id=\"fs-id1169739304017\">[latex]h^{\\prime}(x)=\\frac{-3}{\\sqrt{6x-9x^2}}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739273648\">Find the equation of the line tangent to the graph of [latex]f(x)= \\sin^{-1} x[\/latex] at [latex]x=0[\/latex].<\/p>\r\n<p>[reveal-answer q=\"2340987\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"2340987\"]<\/p>\r\n<p id=\"fs-id1169739299514\">[latex]f^{\\prime}(0)[\/latex] gives the slope of the tangent line.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739302752\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739302752\"]<\/p>\r\n<p id=\"fs-id1169739302752\">[latex]y=x[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>&nbsp;<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Find the derivative of an inverse function<\/li>\n<li>Identify the derivatives for inverse trig functions like arcsine, arccosine, and arctangent<\/li>\n<\/ul>\n<\/section>\n<h2>Derivatives of Various Inverse 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\">Inverse Function Theorem:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For invertible and differentiable function f(x): [latex]\\dfrac{1}{f^{\\prime}(f^{-1}(x))}[\/latex]<\/li>\n<\/ul>\n<\/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\">Tangent lines of [latex]f(x)[\/latex] and [latex]f^(-1)(x)[\/latex] have reciprocal slopes<\/li>\n<li class=\"whitespace-normal break-words\">Symmetric about [latex]y = x[\/latex] line<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Extending the Power Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For positive integer [latex]n[\/latex]: [latex]\\frac{d}{dx}(x^{1\/n}) = \\frac{1}{n}x^{(1\/n)-1}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For positive integer [latex]n[\/latex] and any integer m: [latex]\\frac{d}{dx}(x^{m\/n}) = \\frac{m}{n}x^{(m\/n)-1}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739269403\">Use the inverse function theorem to find the derivative of [latex]g(x)=\\dfrac{1}{x+2}.[\/latex] Compare the result obtained by differentiating [latex]g(x)[\/latex] directly.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q336869\">Show Solution<\/button><\/p>\n<div id=\"q336869\" class=\"hidden-answer\" style=\"display: none\">\n[latex]g^{\\prime}(x)=-\\frac{1}{(x+2)^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739036425\">Use the inverse function theorem to find the derivative of [latex]g(x)=\\sqrt[3]{x}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738969956\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738969956\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738969956\">The function [latex]g(x)=\\sqrt[3]{x}[\/latex] is the inverse of the function [latex]f(x)=x^3[\/latex]. Since [latex]g^{\\prime}(x)=\\frac{1}{f^{\\prime}(g(x))}[\/latex], begin by finding [latex]f^{\\prime}(x)[\/latex]. Thus,<\/p>\n<div id=\"fs-id1169739305067\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=3x^2[\/latex] and [latex]f^{\\prime}(g(x))=3(\\sqrt[3]{x})^2=3x^{2\/3}[\/latex]<\/div>\n<p id=\"fs-id1169739034112\">Finally,<\/p>\n<div id=\"fs-id1169739340252\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]g^{\\prime}(x)=\\frac{1}{3x^{2\/3}}=\\frac{1}{3}x^{-2\/3}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169736607624\">Find the derivative of [latex]s(t)=\\sqrt{2t+1}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q8665521\">Hint<\/button><\/p>\n<div id=\"q8665521\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736596062\">Rewrite as [latex]s(t)=(2t+1)^{1\/2}[\/latex] and use the chain 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-id1169739183503\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739183503\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739183503\">[latex]s^{\\prime}(t)=(2t+1)^{\u22121\/2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Derivatives of Inverse 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\">Key Derivatives:\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}\\frac{d}{dx}(\\sin^{-1} x)=\\large \\frac{1}{\\sqrt{1-x^2}} & & & & \\frac{d}{dx}(\\cos^{-1} x)=\\large \\frac{-1}{\\sqrt{1-x^2}} \\\\ \\frac{d}{dx}(\\tan^{-1} x)=\\large \\frac{1}{1+x^2} & & & & \\frac{d}{dx}(\\cot^{-1} x)=\\large \\frac{-1}{1+x^2} \\\\ \\frac{d}{dx}(\\sec^{-1} x)=\\large \\frac{1}{|x|\\sqrt{x^2-1}} & & & & \\frac{d}{dx}(\\csc^{-1} x)=\\large \\frac{-1}{|x|\\sqrt{x^2-1}} \\end{array}[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">For composite functions like [latex]\\sin^{-1}(g(x))[\/latex], use the chain rule in conjunction with inverse trig derivatives<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to the domains of inverse trigonometric functions when differentiating<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739111210\">Use the inverse function theorem to find the derivative of [latex]g(x)=\\tan^{-1} x[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q462877\">Hint<\/button><\/p>\n<div id=\"q462877\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736613675\">The inverse of [latex]g(x)[\/latex] is [latex]f(x)= \\tan x[\/latex].\u00a0<\/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-id1169736659262\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169736659262\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736659262\">[latex]g^{\\prime}(x)=\\dfrac{1}{1+x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739027843\">Find the derivative of [latex]h(x)=x^2 \\sin^{-1} x[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169736603526\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169736603526\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736603526\">By applying the product rule, we have<\/p>\n<div id=\"fs-id1169736603529\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=2x \\sin^{-1} x+\\frac{1}{\\sqrt{1-x^2}} \\cdot x^2[\/latex].<\/div>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0dlA5QJZYsw?controls=0&amp;start=786&amp;end=848&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\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\/3.7DerivativesOfInverseFuunctions786to848_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.7 Derivatives of Inverse 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-id1169739325671\">Find the derivative of [latex]h(x)= \\cos^{-1} (3x-1)[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739304017\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739304017\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739304017\">[latex]h^{\\prime}(x)=\\frac{-3}{\\sqrt{6x-9x^2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739273648\">Find the equation of the line tangent to the graph of [latex]f(x)= \\sin^{-1} x[\/latex] at [latex]x=0[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q2340987\">Hint<\/button><\/p>\n<div id=\"q2340987\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739299514\">[latex]f^{\\prime}(0)[\/latex] gives the slope of the tangent line.<\/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-id1169739302752\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739302752\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739302752\">[latex]y=x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>&nbsp;<\/p>\n","protected":false},"author":15,"menu_order":18,"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\/3350"}],"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":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3350\/revisions"}],"predecessor-version":[{"id":3781,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3350\/revisions\/3781"}],"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\/3350\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3350"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3350"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3350"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3350"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}