{"id":3315,"date":"2024-06-18T20:15:28","date_gmt":"2024-06-18T20:15:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3315"},"modified":"2024-08-05T01:59:43","modified_gmt":"2024-08-05T01:59:43","slug":"the-chain-rule-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/the-chain-rule-fresh-take\/","title":{"raw":"The Chain Rule: Fresh Take","rendered":"The Chain Rule: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Explain and use the chain rule<\/li>\r\n\t<li>Use the chain rule along with other rules to differentiate functions involving powers, products, quotients, and trigonometry<\/li>\r\n\t<li>Use the chain rule to find derivatives when multiple functions are nested together<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Deriving the Chain Rule<\/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\">Chain Rule Definition:\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]h(x) = f(g(x))[\/latex], the derivative is: [latex]h'(x) = f'(g(x)) \\cdot g'(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Alternative Notation:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">If [latex]y[\/latex] is a function of [latex]u[\/latex], and [latex]u[\/latex] is a function of [latex]x[\/latex]: [latex]\\frac{dy}{dx} = \\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Purpose:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Simplifies differentiation of composite functions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Breaks down complex functions into simpler parts<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Useful for functions like [latex]\\sin(x^3)[\/latex] or [latex]\\sqrt{3x^2+1}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Chain rule can be applied iteratively for functions with multiple compositions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Differentiate [latex]h(x) = e^{\\sin(x^2)}[\/latex] using the chain rule.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><br \/>\r\n[reveal-answer q=\"576268\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"576268\"]<\/p>\r\n<ol>\r\n\t<li class=\"whitespace-normal break-words\">Identify the 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\">Outer function: [latex]f(u) = e^u[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Middle function: [latex]g(v) = \\sin(v)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Inner function: [latex]k(x) = x^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply the chain rule iteratively: [latex]h'(x) = f'(g(k(x))) \\cdot g'(k(x)) \\cdot k'(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Compute each 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\">[latex]f'(u) = e^u[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]g'(v) = \\cos(v)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]k'(x) = 2x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Substitute back into the chain rule: [latex]h'(x) = e^{\\sin(x^2)} \\cdot \\cos(x^2) \\cdot 2x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Simplify: [latex]h'(x) = 2x e^{\\sin(x^2)} \\cos(x^2)[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Combining the Chain Rule With Other Rules<\/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\">Chain Rule with Power Rule:\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]h(x) = (g(x))^n[\/latex], the derivative is: [latex]h'(x) = n(g(x))^{n-1}g'(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Chain Rule with 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\">Example: [latex]\\frac{d}{dx}(\\sin(g(x))) = \\cos(g(x))g'(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Chain Rule with Product Rule:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Apply product rule first, then chain rule to each term<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">General Strategy:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Identify the composition of functions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply appropriate rules in the correct order<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Simplify the result<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169736595961\">Find the derivative of [latex]h(x)=(2x^3+2x-1)^4[\/latex]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739325717\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739325717\"]<\/p>\r\n<p id=\"fs-id1169739325717\">[latex]h^{\\prime}(x)=4(2x^3+2x-1)^3(6x^{2}+2)=8(3x^{2}+1)(2x^3+2x-1)^3[\/latex]<\/p>\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\/CE8oqftYNvQ?controls=0&amp;start=136&amp;end=185&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\/3.6TheChainRule136to185_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.6 The Chain Rule\" 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-id1169739303885\">Find the equation of the line tangent to the graph of [latex]f(x)=(x^2-2)^3[\/latex] at [latex]x=-2[\/latex].<\/p>\r\n<p>[reveal-answer q=\"fs-id1169738869705\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169738869705\"]<\/p>\r\n<p id=\"fs-id1169738869705\">[latex]y=-48x-88[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739188151\">Find the derivative of [latex]h(x)= \\sin (7x+2)[\/latex].<\/p>\r\n<p>[reveal-answer q=\"166577\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"166577\"]<\/p>\r\n<p id=\"fs-id1169736607578\">Apply the chain rule to [latex]h(x)= \\sin g(x)[\/latex] first and then use [latex]g(x)=7x+2[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"232193\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"232193\"]<\/p>\r\n<p>[latex]h^{\\prime}(x)=7 \\cos (7x+2)[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169736603469\">Find the derivative of [latex]h(x)=\\dfrac{x}{(2x+3)^3}[\/latex]<\/p>\r\n<p>[reveal-answer q=\"3778210\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3778210\"]<\/p>\r\n<p id=\"fs-id1169736603573\">Start out by applying the quotient rule. Remember to use the chain rule to differentiate the denominator.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169736603515\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169736603515\"]<\/p>\r\n<p id=\"fs-id1169736603515\">[latex]h^{\\prime}(x)=\\frac{3-4x}{(2x+3)^4}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Applying the Chain Rule Multiple Times<\/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\">Chain Rule for Composition of Three 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\">For [latex]k(x) = h(f(g(x)))[\/latex], the derivative is: [latex]k'(x) = h'(f(g(x))) \\cdot f'(g(x)) \\cdot g'(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">General Approach:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Work from the outside in<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply the chain rule as many times as necessary<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Number of Terms:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">The derivative of a composition of [latex]n[\/latex] functions will have [latex]n[\/latex] terms<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739350751\">Find the derivative of [latex]h(x)=\\sin^6 (x^3)[\/latex]<\/p>\r\n<p>[reveal-answer q=\"9044612\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"9044612\"]<\/p>\r\n<p id=\"fs-id1169739350858\">Rewrite [latex]h(x)=\\sin^6 (x^3)=(\\sin(x^3))^6[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739350792\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739350792\"]<\/p>\r\n<p id=\"fs-id1169739350792\">[latex]h^{\\prime}(x)=18x^2 \\sin^5 (x^3) \\cos (x^3)[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739266719\">A particle moves along a coordinate axis. Its position at time [latex]t[\/latex] is given by [latex]s(t)= \\sin (4t)[\/latex]. Find its acceleration at time [latex]t[\/latex].<\/p>\r\n<p>[reveal-answer q=\"3388209\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3388209\"]<\/p>\r\n<p id=\"fs-id1169736655167\">Acceleration is the second derivative of position.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739266765\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739266765\"]<\/p>\r\n<p id=\"fs-id1169739266765\">[latex]a(t)=-16 \\sin (4t)[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739341448\">Let [latex]h(x)=f(g(x))[\/latex]. If [latex]g(2)=-3, \\, g^{\\prime}(2)=4[\/latex], and [latex]f^{\\prime}(-3)=7[\/latex], find [latex]h^{\\prime}(2)[\/latex].<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739353312\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739353312\"]<\/p>\r\n<p id=\"fs-id1169739353312\">[latex]28[\/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]y = \\ln(\\sin(e^{2x}))[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"215750\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"215750\"]<\/p>\r\n<ol>\r\n\t<li class=\"whitespace-normal break-words\">Identify the layers of composition:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Outermost: [latex]h(u) = \\ln(u)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Middle: [latex]f(v) = \\sin(v)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Innermost: [latex]g(x) = e^{2x}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply the chain rule, starting from the outside: [latex]y' = \\frac{1}{\\sin(e^{2x})} \\cdot \\frac{d}{dx}[\\sin(e^{2x})][\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply the chain rule to [latex]\\sin(e^{2x})[\/latex]: [latex]y' = \\frac{1}{\\sin(e^{2x})} \\cdot \\cos(e^{2x}) \\cdot \\frac{d}{dx}[e^{2x}][\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Differentiate [latex]e^{2x}[\/latex]: [latex]y' = \\frac{1}{\\sin(e^{2x})} \\cdot \\cos(e^{2x}) \\cdot e^{2x} \\cdot 2[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Simplify: [latex]y' = \\frac{2e^{2x} \\cos(e^{2x})}{\\sin(e^{2x})}[\/latex]<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>The Chain Rule Using Leibniz\u2019s Notation<\/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\">Leibniz's Notation for Chain Rule:\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]y = f(u)[\/latex] and [latex]u = g(x)[\/latex], the chain rule is expressed as: [latex]\\frac{dy}{dx} = \\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">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\">[latex]\\frac{dy}{dx}[\/latex]: Rate of change of [latex]y[\/latex] with respect to [latex]x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{dy}{du}[\/latex]: Rate of change of [latex]y[\/latex] with respect to [latex]u[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{du}{dx}[\/latex]: Rate of change of [latex]u[\/latex] with respect to [latex]x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Advantages:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Provides a clear visual representation of the chain rule<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Helps in understanding the relationship between variables<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739302090\">Use Leibniz\u2019s notation to find the derivative of [latex]y= \\cos (x^3)[\/latex]. Make sure that the final answer is expressed entirely in terms of the variable [latex]x[\/latex].<\/p>\r\n<p>[reveal-answer q=\"844612\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"844612\"]<\/p>\r\n<p id=\"fs-id1169736594079\">Let [latex]u=x^3[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169736594031\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169736594031\"]<\/p>\r\n<p id=\"fs-id1169736594031\">[latex]\\frac{dy}{dx}=-3x^2 \\sin (x^3)[\/latex]<\/p>\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\/CE8oqftYNvQ?controls=0&amp;start=1155&amp;end=1203&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\/3.6TheChainRule1155to1203_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.6 The Chain Rule\" 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]y = \\sqrt{\\sin(2x^3)}[\/latex] using Leibniz's notation.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><br \/>\r\n[reveal-answer q=\"301323\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"301323\"]<\/p>\r\n<p>Define intermediate variables:<\/p>\r\n<p>Let [latex]u = \\sin(2x^3)[\/latex] and [latex]v = 2x^3[\/latex]<\/p>\r\n<p>Express [latex]y[\/latex] in terms of [latex]u[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]y = \\sqrt{u} = u^{\\frac{1}{2}}[\/latex]<\/p>\r\n<p>Find [latex]\\frac{dy}{du}[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{dy}{du} = \\frac{1}{2}u^{-\\frac{1}{2}} = \\frac{1}{2\\sqrt{u}}[\/latex]<\/p>\r\n<p>Find [latex]\\frac{du}{dv}[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{du}{dv} = \\cos(v)[\/latex]<\/p>\r\n<p>Find [latex]\\frac{dv}{dx}[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{dv}{dx} = 6x^2[\/latex]<\/p>\r\n<p>Apply the chain rule:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{dy}{dx} = \\frac{dy}{du} \\cdot \\frac{du}{dv} \\cdot \\frac{dv}{dx}[\/latex]<\/p>\r\n<p>Substitute and simplify:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}<br \/>\r\n\\frac{dy}{dx} &amp;=&amp; \\dfrac{1}{2\\sqrt{u}} \\cdot \\cos(v) \\cdot 6x^2 \\\\<br \/>\r\n&amp;=&amp; \\dfrac{1}{2\\sqrt{\\sin(2x^3)}} \\cdot \\cos(2x^3) \\cdot 6x^2 \\\\<br \/>\r\n&amp;=&amp; \\dfrac{3x^2 \\cos(2x^3)}{\\sqrt{\\sin(2x^3)}}<br \/>\r\n\\end{array}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Explain and use the chain rule<\/li>\n<li>Use the chain rule along with other rules to differentiate functions involving powers, products, quotients, and trigonometry<\/li>\n<li>Use the chain rule to find derivatives when multiple functions are nested together<\/li>\n<\/ul>\n<\/section>\n<h2>Deriving the Chain Rule<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Chain Rule Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]h(x) = f(g(x))[\/latex], the derivative is: [latex]h'(x) = f'(g(x)) \\cdot g'(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Alternative Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]y[\/latex] is a function of [latex]u[\/latex], and [latex]u[\/latex] is a function of [latex]x[\/latex]: [latex]\\frac{dy}{dx} = \\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Purpose:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Simplifies differentiation of composite functions<\/li>\n<li class=\"whitespace-normal break-words\">Breaks down complex functions into simpler parts<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Useful for functions like [latex]\\sin(x^3)[\/latex] or [latex]\\sqrt{3x^2+1}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Chain rule can be applied iteratively for functions with multiple compositions<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Differentiate [latex]h(x) = e^{\\sin(x^2)}[\/latex] using the chain rule.<\/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=\"q576268\">Show Answer<\/button><\/p>\n<div id=\"q576268\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Identify the functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Outer function: [latex]f(u) = e^u[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Middle function: [latex]g(v) = \\sin(v)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Inner function: [latex]k(x) = x^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Apply the chain rule iteratively: [latex]h'(x) = f'(g(k(x))) \\cdot g'(k(x)) \\cdot k'(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Compute each derivative:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f'(u) = e^u[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]g'(v) = \\cos(v)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]k'(x) = 2x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Substitute back into the chain rule: [latex]h'(x) = e^{\\sin(x^2)} \\cdot \\cos(x^2) \\cdot 2x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify: [latex]h'(x) = 2x e^{\\sin(x^2)} \\cos(x^2)[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<h2>Combining the Chain Rule With Other Rules<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Chain Rule with Power Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]h(x) = (g(x))^n[\/latex], the derivative is: [latex]h'(x) = n(g(x))^{n-1}g'(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Chain Rule with Trigonometric Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Example: [latex]\\frac{d}{dx}(\\sin(g(x))) = \\cos(g(x))g'(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Chain Rule with Product Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Apply product rule first, then chain rule to each term<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Strategy:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the composition of functions<\/li>\n<li class=\"whitespace-normal break-words\">Apply appropriate rules in the correct order<\/li>\n<li class=\"whitespace-normal break-words\">Simplify the result<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169736595961\">Find the derivative of [latex]h(x)=(2x^3+2x-1)^4[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739325717\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739325717\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739325717\">[latex]h^{\\prime}(x)=4(2x^3+2x-1)^3(6x^{2}+2)=8(3x^{2}+1)(2x^3+2x-1)^3[\/latex]<\/p>\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\/CE8oqftYNvQ?controls=0&amp;start=136&amp;end=185&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\/3.6TheChainRule136to185_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.6 The Chain Rule&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739303885\">Find the equation of the line tangent to the graph of [latex]f(x)=(x^2-2)^3[\/latex] at [latex]x=-2[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738869705\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738869705\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738869705\">[latex]y=-48x-88[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739188151\">Find the derivative of [latex]h(x)= \\sin (7x+2)[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q166577\">Hint<\/button><\/p>\n<div id=\"q166577\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736607578\">Apply the chain rule to [latex]h(x)= \\sin g(x)[\/latex] first and then use [latex]g(x)=7x+2[\/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=\"q232193\">Show Solution<\/button><\/p>\n<div id=\"q232193\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]h^{\\prime}(x)=7 \\cos (7x+2)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169736603469\">Find the derivative of [latex]h(x)=\\dfrac{x}{(2x+3)^3}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3778210\">Hint<\/button><\/p>\n<div id=\"q3778210\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736603573\">Start out by applying the quotient rule. Remember to use the chain rule to differentiate the denominator.<\/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-id1169736603515\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169736603515\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736603515\">[latex]h^{\\prime}(x)=\\frac{3-4x}{(2x+3)^4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Applying the Chain Rule Multiple Times<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Chain Rule for Composition of Three Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]k(x) = h(f(g(x)))[\/latex], the derivative is: [latex]k'(x) = h'(f(g(x))) \\cdot f'(g(x)) \\cdot g'(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Approach:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Work from the outside in<\/li>\n<li class=\"whitespace-normal break-words\">Apply the chain rule as many times as necessary<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Number of Terms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The derivative of a composition of [latex]n[\/latex] functions will have [latex]n[\/latex] terms<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739350751\">Find the derivative of [latex]h(x)=\\sin^6 (x^3)[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q9044612\">Hint<\/button><\/p>\n<div id=\"q9044612\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739350858\">Rewrite [latex]h(x)=\\sin^6 (x^3)=(\\sin(x^3))^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-id1169739350792\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739350792\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739350792\">[latex]h^{\\prime}(x)=18x^2 \\sin^5 (x^3) \\cos (x^3)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739266719\">A particle moves along a coordinate axis. Its position at time [latex]t[\/latex] is given by [latex]s(t)= \\sin (4t)[\/latex]. Find its acceleration at time [latex]t[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3388209\">Hint<\/button><\/p>\n<div id=\"q3388209\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736655167\">Acceleration is the second derivative of position.<\/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-id1169739266765\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739266765\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739266765\">[latex]a(t)=-16 \\sin (4t)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739341448\">Let [latex]h(x)=f(g(x))[\/latex]. If [latex]g(2)=-3, \\, g^{\\prime}(2)=4[\/latex], and [latex]f^{\\prime}(-3)=7[\/latex], find [latex]h^{\\prime}(2)[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739353312\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739353312\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739353312\">[latex]28[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find the derivative of [latex]y = \\ln(\\sin(e^{2x}))[\/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=\"q215750\">Show Answer<\/button><\/p>\n<div id=\"q215750\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Identify the layers of composition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Outermost: [latex]h(u) = \\ln(u)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Middle: [latex]f(v) = \\sin(v)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Innermost: [latex]g(x) = e^{2x}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Apply the chain rule, starting from the outside: [latex]y' = \\frac{1}{\\sin(e^{2x})} \\cdot \\frac{d}{dx}[\\sin(e^{2x})][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Apply the chain rule to [latex]\\sin(e^{2x})[\/latex]: [latex]y' = \\frac{1}{\\sin(e^{2x})} \\cdot \\cos(e^{2x}) \\cdot \\frac{d}{dx}[e^{2x}][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Differentiate [latex]e^{2x}[\/latex]: [latex]y' = \\frac{1}{\\sin(e^{2x})} \\cdot \\cos(e^{2x}) \\cdot e^{2x} \\cdot 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify: [latex]y' = \\frac{2e^{2x} \\cos(e^{2x})}{\\sin(e^{2x})}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>The Chain Rule Using Leibniz\u2019s Notation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Leibniz&#8217;s Notation for Chain Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]y = f(u)[\/latex] and [latex]u = g(x)[\/latex], the chain rule is expressed as: [latex]\\frac{dy}{dx} = \\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{dy}{dx}[\/latex]: Rate of change of [latex]y[\/latex] with respect to [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\frac{dy}{du}[\/latex]: Rate of change of [latex]y[\/latex] with respect to [latex]u[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\frac{du}{dx}[\/latex]: Rate of change of [latex]u[\/latex] with respect to [latex]x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Advantages:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Provides a clear visual representation of the chain rule<\/li>\n<li class=\"whitespace-normal break-words\">Helps in understanding the relationship between variables<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739302090\">Use Leibniz\u2019s notation to find the derivative of [latex]y= \\cos (x^3)[\/latex]. Make sure that the final answer is expressed entirely in terms of the variable [latex]x[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q844612\">Hint<\/button><\/p>\n<div id=\"q844612\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736594079\">Let [latex]u=x^3[\/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-id1169736594031\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169736594031\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736594031\">[latex]\\frac{dy}{dx}=-3x^2 \\sin (x^3)[\/latex]<\/p>\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\/CE8oqftYNvQ?controls=0&amp;start=1155&amp;end=1203&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\/3.6TheChainRule1155to1203_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.6 The Chain Rule&#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]y = \\sqrt{\\sin(2x^3)}[\/latex] using Leibniz&#8217;s notation.<\/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=\"q301323\">Show Answer<\/button><\/p>\n<div id=\"q301323\" class=\"hidden-answer\" style=\"display: none\">\n<p>Define intermediate variables:<\/p>\n<p>Let [latex]u = \\sin(2x^3)[\/latex] and [latex]v = 2x^3[\/latex]<\/p>\n<p>Express [latex]y[\/latex] in terms of [latex]u[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]y = \\sqrt{u} = u^{\\frac{1}{2}}[\/latex]<\/p>\n<p>Find [latex]\\frac{dy}{du}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{dy}{du} = \\frac{1}{2}u^{-\\frac{1}{2}} = \\frac{1}{2\\sqrt{u}}[\/latex]<\/p>\n<p>Find [latex]\\frac{du}{dv}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{du}{dv} = \\cos(v)[\/latex]<\/p>\n<p>Find [latex]\\frac{dv}{dx}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{dv}{dx} = 6x^2[\/latex]<\/p>\n<p>Apply the chain rule:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{dy}{dx} = \\frac{dy}{du} \\cdot \\frac{du}{dv} \\cdot \\frac{dv}{dx}[\/latex]<\/p>\n<p>Substitute and simplify:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}<br \/>  \\frac{dy}{dx} &=& \\dfrac{1}{2\\sqrt{u}} \\cdot \\cos(v) \\cdot 6x^2 \\\\<br \/>  &=& \\dfrac{1}{2\\sqrt{\\sin(2x^3)}} \\cdot \\cos(2x^3) \\cdot 6x^2 \\\\<br \/>  &=& \\dfrac{3x^2 \\cos(2x^3)}{\\sqrt{\\sin(2x^3)}}<br \/>  \\end{array}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":14,"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\/3315"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3315\/revisions"}],"predecessor-version":[{"id":3778,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3315\/revisions\/3778"}],"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\/3315\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3315"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3315"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3315"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3315"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}