{"id":601,"date":"2025-10-09T15:24:30","date_gmt":"2025-10-09T15:24:30","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=601"},"modified":"2025-10-29T14:14:02","modified_gmt":"2025-10-29T14:14:02","slug":"introduction-to-partial-derivatives-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/introduction-to-partial-derivatives-fresh-take\/","title":{"raw":"Introduction to Partial Derivatives: Fresh Take","rendered":"Introduction to Partial Derivatives: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Calculate the partial derivatives of a function of two variables<\/li>\r\n \t<li>Calculate the partial derivatives of a function of more than two variables<\/li>\r\n \t<li>Find higher-order partial derivatives of a function of two variables<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Derivatives of a Function of Two Variables<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">With two independent variables, we can't just take \"the\" derivative. Instead, we take <strong>partial derivatives<\/strong>\u2014one with respect to each variable\u2014by holding the other variable constant.<\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]z = f(x, y)[\/latex], we have two partial derivatives:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial x}[\/latex]: measures how [latex]f[\/latex] changes as [latex]x[\/latex] changes (holding [latex]y[\/latex] fixed)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial y}[\/latex]: measures how [latex]f[\/latex] changes as [latex]y[\/latex] changes (holding [latex]x[\/latex] fixed)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>How to Calculate<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Treat the other variable as a constant and differentiate normally.<\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]\\dfrac{\\partial f}{\\partial x}[\/latex]: Treat [latex]y[\/latex] as constant, differentiate with respect to [latex]x[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]\\dfrac{\\partial f}{\\partial y}[\/latex]: Treat [latex]x[\/latex] as constant, differentiate with respect to [latex]y[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Geometric Meaning<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial x}[\/latex]: slope of the tangent line parallel to the [latex]x[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial y}[\/latex]: slope of the tangent line parallel to the [latex]y[\/latex]-axis<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Think of standing on the surface and walking parallel to the [latex]x[\/latex]-axis. The steepness you feel is [latex]\\dfrac{\\partial f}{\\partial x}[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>All the Old Rules Apply<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Power rule, product rule, quotient rule, chain rule\u2014everything from single-variable calculus works. Just remember which variable you're differentiating with respect to.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use the definition of the partial derivative as a limit to calculate [latex]\\partial f\/\\partial x[\/latex] and [latex]\\partial f\/\\partial y[\/latex] for the function\r\n<p style=\"text-align: center;\">[latex]f\\,(x,\\ y)=4x^{2}+2xy-y^{2}+3x-2y+5.[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1367793933124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1367793933124\"]\r\n<div style=\"text-align: center;\">[latex]\\frac{\\partial f}{\\partial x}=8x+2y+3,\\ \\frac{\\partial f}{\\partial y}=2x-2y-2[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Calculate [latex]\\partial f\/\\partial x[\/latex] and [latex]\\partial f\/\\partial y[\/latex] for the function [latex]f\\,(x,\\ y)=\\tan{(x^{3}-3x^{2}y^{2}+2y^{4})}[\/latex] by holding the opposite variable constant, then differentiating.[reveal-answer q=\"fs-id1367703933124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1367703933124\"]\r\n<div style=\"text-align: center;\">[latex]\\frac{\\partial f}{\\partial x}=(3x^{2}-6xy^{2})\\sec^{2}{(x^{3}-3x^{2}y^{2}+2y^{4})}[\/latex]<\/div>\r\n&nbsp;\r\n<div style=\"text-align: center;\">[latex]\\frac{\\partial f}{\\partial y}=(-6x^{2}y+8y^{3})\\sec^{2}{(x^{3}-3x^{2}y^{2}+2y^{4})}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186153&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=LsVhLEj4oP8&amp;video_target=tpm-plugin-9m34t2ja-LsVhLEj4oP8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.13_transcript.html\">transcript for \u201cCP 4.13\u201d here (opens in new window).<\/a><\/center><\/section><section class=\"textbox example\" aria-label=\"Example\">Use a contour map to estimate [latex]\\partial f\/\\partial y[\/latex] at point [latex](0,\\ \\sqrt{2})[\/latex] for the function\r\n<p style=\"text-align: center;\">[latex]f\\,(x,\\ y)=x^{2}-y^{2}[\/latex].<\/p>\r\nCompare this with the exact answer.\r\n\r\n[reveal-answer q=\"fs-id1367703938124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1367703938124\"]\r\n<p style=\"text-align: left;\">Using the curves corresponding to [latex]c=-2[\/latex] and [latex]c=-3[\/latex], we obtain<\/p>\r\n\r\n<div style=\"text-align: center;\">[latex]\\large{\\frac{\\partial f}{\\partial y}\\bigg|_{(x,\\ y)=(0,\\ \\sqrt{2})}\\approx\\frac{f\\,(0,\\ \\sqrt{3})-f\\,(0,\\ \\sqrt{2})}{\\sqrt{3}-\\sqrt{2}}=\\frac{-3-(-2)}{\\sqrt{3}-\\sqrt{2}}\\cdot\\frac{\\sqrt{3}+\\sqrt{2}}{\\sqrt{3}+\\sqrt{2}}=-\\sqrt{3}-\\sqrt{2}\\approx-3.146}[\/latex].<\/div>\r\n&nbsp;\r\n\r\nThe exact answer is\r\n<div style=\"text-align: center;\">[latex]\\large{\\frac{\\partial f}{\\partial y}\\bigg|_{(x,\\ y)(0,\\ \\sqrt{2}}=-2y\\big|_{(x,\\ y)=(0,\\ \\sqrt{2})}=-2\\sqrt{2}\\approx -2.828.}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Functions of More Than Two Variables<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Partial derivatives extend naturally to three or more variables. The principle stays the same: hold all other variables constant and differentiate with respect to one variable at a time.<\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]w = f(x, y, z)[\/latex], we have three partial derivatives:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial x}[\/latex]: hold [latex]y[\/latex] and [latex]z[\/latex] constant, differentiate with respect to [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial y}[\/latex]: hold [latex]x[\/latex] and [latex]z[\/latex] constant, differentiate with respect to [latex]y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial z}[\/latex]: hold [latex]x[\/latex] and [latex]y[\/latex] constant, differentiate with respect to [latex]z[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The number of partial derivatives equals the number of independent variables. Four variables? Four partial derivatives. Five variables? Five partial derivatives.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Identify which variable you're differentiating with respect to. Treat all other variables as constants. Apply your standard differentiation rules.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use the limit definition of partial derivatives to calculate [latex]\\partial f\/\\partial x[\/latex] for the function\r\n<p style=\"text-align: center;\">[latex]f\\,(x,\\ y,\\ z)=2x^{2}-4x^{2}+2y^{2}+5xz^{2}-6x+3z-8[\/latex].<\/p>\r\nThen, find [latex]\\partial f\/\\partial y[\/latex] and [latex]\\partial f\/\\partial z[\/latex] by setting the other two variables constant and differentiating accordingly.\r\n\r\n[reveal-answer q=\"fs-id1367703930124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1367703930124\"]\r\n\r\n[latex] \\begin{alignat}{2} \\hspace{5cm} \\frac{\\partial f}{\\partial x}&amp;=4x-8xy+5z^{2}-6 \\\\\r\n\r\n\\frac{\\partial f}{\\partial y}&amp; = -4x^{2}+4y&amp;\\quad\\\\\r\n\r\n\\frac{\\partial f}{\\partial z}&amp; =10xz+3\\\\\r\n\r\n\\end{alignat}[\/latex]\r\n<div style=\"text-align: left;\"><\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Calculate [latex]\\partial f\/\\partial x,\\ \\partial f\/\\partial y[\/latex], and [latex]\\partial f\/\\partial z[\/latex] for the function [latex]f\\,(x,\\ y,\\ z)=\\sec{(x^{2}y)}-\\tan{(x^{3}yz^{2}})[\/latex].[reveal-answer q=\"fs-id1367773930124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1367773930124\"]\r\n\r\n[latex] \\begin{alignat}{2} \\hspace{5cm} \\frac{\\partial f}{\\partial x}&amp;=2xy\\sec{(x^{2}y)}\\tan{(x^{2}y)}-3x^{2}yz^{2}\\sec^{2}{(x^{3}yz^{2})} \\\\\r\n\r\n\\frac{\\partial f}{\\partial y}&amp;=x^{2}\\sec{(x^{2}y)}\\tan{(x^{2}y)-x^{3}z^{2}}\\sec^{2}{(x^{3}yz^{2})}&amp;\\quad\\\\\r\n\r\n\\frac{\\partial f}{\\partial z}&amp;=-2x^{3}yz\\sec^{2}{(x^{3}yz^{2})}\\\\\r\n\r\n\\end{alignat}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186154&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=cp5oduxbhuA&amp;video_target=tpm-plugin-9qc3yru5-cp5oduxbhuA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.16_transcript.html\">transcript for \u201cCP 4.16\u201d here (opens in new window).<\/a><\/center><\/section>\r\n<h2>Higher-Order Partial Derivatives<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Just as you can take second, third, and higher derivatives in single-variable calculus, you can take higher-order partial derivatives. With two variables, there are four second-order partial derivatives.<\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]f(x, y)[\/latex]:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial^2 f}{\\partial x^2} = f_{xx}[\/latex]: differentiate twice with respect to [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial^2 f}{\\partial y^2} = f_{yy}[\/latex]: differentiate twice with respect to [latex]y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial^2 f}{\\partial x \\partial y} = f_{xy}[\/latex]: differentiate first with respect to [latex]x[\/latex], then [latex]y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial^2 f}{\\partial y \\partial x} = f_{yx}[\/latex]: differentiate first with respect to [latex]y[\/latex], then [latex]x[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The last two are called <strong>mixed partial derivatives<\/strong> because they involve differentiating with respect to different variables.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Reading the Notation<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">In [latex]\\dfrac{\\partial^2 f}{\\partial y \\partial x}[\/latex]: read right to left (differentiate [latex]x[\/latex] first, then [latex]y[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">In [latex]f_{xy}[\/latex]: read left to right (differentiate [latex]x[\/latex] first, then [latex]y[\/latex])<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Clairaut's Theorem<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">If the mixed partial derivatives are continuous, then [latex]f_{xy} = f_{yx}[\/latex]. The order doesn't matter.<\/p>\r\n<p class=\"whitespace-normal break-words\">This holds for most functions you'll encounter. It means you can choose whichever order makes the calculation easier.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Find the first partial derivatives. Then differentiate each one with respect to both variables. Check that the mixed partials match\u2014they should if the function is well-behaved.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Calculate all four second partial derivatives for the function\r\n<div style=\"text-align: center;\">[latex]\\large{f\\,(x,\\ y)=\\sin{(3x-2y)}+\\cos{(x+4y)}.}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n[reveal-answer q=\"fs-id1567773930324\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1567773930324\"]\r\n<div style=\"text-align: center;\">[latex]\\frac{\\partial^{2}f}{\\partial x^{2}}=-9\\sin{(3x-2y)}-\\cos{(x+4y)}[\/latex]<\/div>\r\n&nbsp;\r\n<div style=\"text-align: center;\">[latex]\\frac{\\partial^{2}f}{\\partial x\\partial y}=6\\sin{(3x-2y)}-4\\cos{(x+4y)}[\/latex]<\/div>\r\n&nbsp;\r\n<div style=\"text-align: center;\">[latex]\\frac{\\partial^{2}f}{\\partial y\\partial x}=6\\sin{(3x-2y)}-4\\cos{(x+4y)}[\/latex]<\/div>\r\n&nbsp;\r\n<div style=\"text-align: center;\">[latex]\\frac{\\partial^{2}f}{\\partial y^{2}}=-4\\sin{(3x-2y)}-16\\cos{(x+4y)}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186155&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=lIhClFtXppQ&amp;video_target=tpm-plugin-k9q1ge88-lIhClFtXppQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.17_transcript.html\"> transcript for \u201cCP 4.17\u201d here (opens in new window).<\/a><\/center><\/section>\r\n<h2>Partial Differential Equations<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">A <strong>partial differential equation<\/strong> (PDE) involves an unknown function of multiple variables and its partial derivatives. These equations model physical phenomena like heat flow, wave propagation, and fluid dynamics.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Common Examples<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Heat equation<\/strong>: [latex]u_t = c^2(u_{xx} + u_{yy})[\/latex] describes temperature distribution over time<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Wave equation<\/strong>: [latex]u_{tt} = c^2(u_{xx} + u_{yy})[\/latex] describes vibrations and wave motion<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Laplace's equation<\/strong>: [latex]u_{xx} + u_{yy} = 0[\/latex] describes steady-state phenomena<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">In these equations, [latex]x[\/latex] and [latex]y[\/latex] represent spatial coordinates, and [latex]t[\/latex] represents time.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Verifying Solutions<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">To verify a proposed solution:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Calculate all required partial derivatives<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute them into the PDE<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify to check if both sides match<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>Visualizing Solutions<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Solutions to PDEs are functions of multiple variables. For the heat equation with time, we can graph snapshots at different times to see how temperature evolves. As time increases, extreme temperatures level out.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Calculate the needed partial derivatives carefully. Substitute into the PDE. Simplify both sides. Verify equality.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Verify that [latex]u\\,(x,\\ y,\\ t)=2\\,\\sin\\,(\\frac{x}{3})\\,\\sin\\,(\\frac{y}{4})\\,e^{-25t\/16}[\/latex] is a solution to the heat equation\r\n<p style=\"text-align: center;\">[latex]u_{t}=9(u_{xx}+u_{yy})[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1367573930124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1367573930124\"]\r\n<div style=\"text-align: center;\"><\/div>\r\n[latex]u\\,(x,\\ y,\\ t)=2\\,\\sin\\,(\\frac{x}{3})\\,\\sin\\,(\\frac{y}{4})\\,e^{-25t\/16}[\/latex] is a solution to the heat equation. See worked example video below to verify the solution.\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186156&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=pMoH9Key-B8&amp;video_target=tpm-plugin-k0fmirvu-pMoH9Key-B8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.18_transcript.html\">transcript for \u201cCP 4.18\u201d here (opens in new window).<\/a><\/center><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Calculate the partial derivatives of a function of two variables<\/li>\n<li>Calculate the partial derivatives of a function of more than two variables<\/li>\n<li>Find higher-order partial derivatives of a function of two variables<\/li>\n<\/ul>\n<\/section>\n<h2>Derivatives of a Function of Two Variables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">With two independent variables, we can&#8217;t just take &#8220;the&#8221; derivative. Instead, we take <strong>partial derivatives<\/strong>\u2014one with respect to each variable\u2014by holding the other variable constant.<\/p>\n<p class=\"whitespace-normal break-words\">For [latex]z = f(x, y)[\/latex], we have two partial derivatives:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial x}[\/latex]: measures how [latex]f[\/latex] changes as [latex]x[\/latex] changes (holding [latex]y[\/latex] fixed)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial y}[\/latex]: measures how [latex]f[\/latex] changes as [latex]y[\/latex] changes (holding [latex]x[\/latex] fixed)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>How to Calculate<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Treat the other variable as a constant and differentiate normally.<\/p>\n<p class=\"whitespace-normal break-words\">For [latex]\\dfrac{\\partial f}{\\partial x}[\/latex]: Treat [latex]y[\/latex] as constant, differentiate with respect to [latex]x[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">For [latex]\\dfrac{\\partial f}{\\partial y}[\/latex]: Treat [latex]x[\/latex] as constant, differentiate with respect to [latex]y[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Geometric Meaning<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial x}[\/latex]: slope of the tangent line parallel to the [latex]x[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial y}[\/latex]: slope of the tangent line parallel to the [latex]y[\/latex]-axis<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Think of standing on the surface and walking parallel to the [latex]x[\/latex]-axis. The steepness you feel is [latex]\\dfrac{\\partial f}{\\partial x}[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>All the Old Rules Apply<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Power rule, product rule, quotient rule, chain rule\u2014everything from single-variable calculus works. Just remember which variable you&#8217;re differentiating with respect to.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use the definition of the partial derivative as a limit to calculate [latex]\\partial f\/\\partial x[\/latex] and [latex]\\partial f\/\\partial y[\/latex] for the function<\/p>\n<p style=\"text-align: center;\">[latex]f\\,(x,\\ y)=4x^{2}+2xy-y^{2}+3x-2y+5.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1367793933124\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1367793933124\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\frac{\\partial f}{\\partial x}=8x+2y+3,\\ \\frac{\\partial f}{\\partial y}=2x-2y-2[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Calculate [latex]\\partial f\/\\partial x[\/latex] and [latex]\\partial f\/\\partial y[\/latex] for the function [latex]f\\,(x,\\ y)=\\tan{(x^{3}-3x^{2}y^{2}+2y^{4})}[\/latex] by holding the opposite variable constant, then differentiating.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1367703933124\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1367703933124\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\frac{\\partial f}{\\partial x}=(3x^{2}-6xy^{2})\\sec^{2}{(x^{3}-3x^{2}y^{2}+2y^{4})}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{\\partial f}{\\partial y}=(-6x^{2}y+8y^{3})\\sec^{2}{(x^{3}-3x^{2}y^{2}+2y^{4})}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186153&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=LsVhLEj4oP8&amp;video_target=tpm-plugin-9m34t2ja-LsVhLEj4oP8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.13_transcript.html\">transcript for \u201cCP 4.13\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Use a contour map to estimate [latex]\\partial f\/\\partial y[\/latex] at point [latex](0,\\ \\sqrt{2})[\/latex] for the function<\/p>\n<p style=\"text-align: center;\">[latex]f\\,(x,\\ y)=x^{2}-y^{2}[\/latex].<\/p>\n<p>Compare this with the exact answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1367703938124\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1367703938124\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">Using the curves corresponding to [latex]c=-2[\/latex] and [latex]c=-3[\/latex], we obtain<\/p>\n<div style=\"text-align: center;\">[latex]\\large{\\frac{\\partial f}{\\partial y}\\bigg|_{(x,\\ y)=(0,\\ \\sqrt{2})}\\approx\\frac{f\\,(0,\\ \\sqrt{3})-f\\,(0,\\ \\sqrt{2})}{\\sqrt{3}-\\sqrt{2}}=\\frac{-3-(-2)}{\\sqrt{3}-\\sqrt{2}}\\cdot\\frac{\\sqrt{3}+\\sqrt{2}}{\\sqrt{3}+\\sqrt{2}}=-\\sqrt{3}-\\sqrt{2}\\approx-3.146}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p>The exact answer is<\/p>\n<div style=\"text-align: center;\">[latex]\\large{\\frac{\\partial f}{\\partial y}\\bigg|_{(x,\\ y)(0,\\ \\sqrt{2}}=-2y\\big|_{(x,\\ y)=(0,\\ \\sqrt{2})}=-2\\sqrt{2}\\approx -2.828.}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>Functions of More Than Two Variables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Partial derivatives extend naturally to three or more variables. The principle stays the same: hold all other variables constant and differentiate with respect to one variable at a time.<\/p>\n<p class=\"whitespace-normal break-words\">For [latex]w = f(x, y, z)[\/latex], we have three partial derivatives:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial x}[\/latex]: hold [latex]y[\/latex] and [latex]z[\/latex] constant, differentiate with respect to [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial y}[\/latex]: hold [latex]x[\/latex] and [latex]z[\/latex] constant, differentiate with respect to [latex]y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial f}{\\partial z}[\/latex]: hold [latex]x[\/latex] and [latex]y[\/latex] constant, differentiate with respect to [latex]z[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The number of partial derivatives equals the number of independent variables. Four variables? Four partial derivatives. Five variables? Five partial derivatives.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Identify which variable you&#8217;re differentiating with respect to. Treat all other variables as constants. Apply your standard differentiation rules.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use the limit definition of partial derivatives to calculate [latex]\\partial f\/\\partial x[\/latex] for the function<\/p>\n<p style=\"text-align: center;\">[latex]f\\,(x,\\ y,\\ z)=2x^{2}-4x^{2}+2y^{2}+5xz^{2}-6x+3z-8[\/latex].<\/p>\n<p>Then, find [latex]\\partial f\/\\partial y[\/latex] and [latex]\\partial f\/\\partial z[\/latex] by setting the other two variables constant and differentiating accordingly.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1367703930124\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1367703930124\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{alignat}{2} \\hspace{5cm} \\frac{\\partial f}{\\partial x}&=4x-8xy+5z^{2}-6 \\\\    \\frac{\\partial f}{\\partial y}& = -4x^{2}+4y&\\quad\\\\    \\frac{\\partial f}{\\partial z}& =10xz+3\\\\    \\end{alignat}[\/latex]<\/p>\n<div style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Calculate [latex]\\partial f\/\\partial x,\\ \\partial f\/\\partial y[\/latex], and [latex]\\partial f\/\\partial z[\/latex] for the function [latex]f\\,(x,\\ y,\\ z)=\\sec{(x^{2}y)}-\\tan{(x^{3}yz^{2}})[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1367773930124\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1367773930124\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{alignat}{2} \\hspace{5cm} \\frac{\\partial f}{\\partial x}&=2xy\\sec{(x^{2}y)}\\tan{(x^{2}y)}-3x^{2}yz^{2}\\sec^{2}{(x^{3}yz^{2})} \\\\    \\frac{\\partial f}{\\partial y}&=x^{2}\\sec{(x^{2}y)}\\tan{(x^{2}y)-x^{3}z^{2}}\\sec^{2}{(x^{3}yz^{2})}&\\quad\\\\    \\frac{\\partial f}{\\partial z}&=-2x^{3}yz\\sec^{2}{(x^{3}yz^{2})}\\\\    \\end{alignat}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186154&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=cp5oduxbhuA&amp;video_target=tpm-plugin-9qc3yru5-cp5oduxbhuA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.16_transcript.html\">transcript for \u201cCP 4.16\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<h2>Higher-Order Partial Derivatives<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Just as you can take second, third, and higher derivatives in single-variable calculus, you can take higher-order partial derivatives. With two variables, there are four second-order partial derivatives.<\/p>\n<p class=\"whitespace-normal break-words\">For [latex]f(x, y)[\/latex]:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial^2 f}{\\partial x^2} = f_{xx}[\/latex]: differentiate twice with respect to [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial^2 f}{\\partial y^2} = f_{yy}[\/latex]: differentiate twice with respect to [latex]y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial^2 f}{\\partial x \\partial y} = f_{xy}[\/latex]: differentiate first with respect to [latex]x[\/latex], then [latex]y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\partial^2 f}{\\partial y \\partial x} = f_{yx}[\/latex]: differentiate first with respect to [latex]y[\/latex], then [latex]x[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The last two are called <strong>mixed partial derivatives<\/strong> because they involve differentiating with respect to different variables.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Reading the Notation<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">In [latex]\\dfrac{\\partial^2 f}{\\partial y \\partial x}[\/latex]: read right to left (differentiate [latex]x[\/latex] first, then [latex]y[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">In [latex]f_{xy}[\/latex]: read left to right (differentiate [latex]x[\/latex] first, then [latex]y[\/latex])<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Clairaut&#8217;s Theorem<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">If the mixed partial derivatives are continuous, then [latex]f_{xy} = f_{yx}[\/latex]. The order doesn&#8217;t matter.<\/p>\n<p class=\"whitespace-normal break-words\">This holds for most functions you&#8217;ll encounter. It means you can choose whichever order makes the calculation easier.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Find the first partial derivatives. Then differentiate each one with respect to both variables. Check that the mixed partials match\u2014they should if the function is well-behaved.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Calculate all four second partial derivatives for the function<\/p>\n<div style=\"text-align: center;\">[latex]\\large{f\\,(x,\\ y)=\\sin{(3x-2y)}+\\cos{(x+4y)}.}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1567773930324\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1567773930324\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\frac{\\partial^{2}f}{\\partial x^{2}}=-9\\sin{(3x-2y)}-\\cos{(x+4y)}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{\\partial^{2}f}{\\partial x\\partial y}=6\\sin{(3x-2y)}-4\\cos{(x+4y)}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{\\partial^{2}f}{\\partial y\\partial x}=6\\sin{(3x-2y)}-4\\cos{(x+4y)}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{\\partial^{2}f}{\\partial y^{2}}=-4\\sin{(3x-2y)}-16\\cos{(x+4y)}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186155&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=lIhClFtXppQ&amp;video_target=tpm-plugin-k9q1ge88-lIhClFtXppQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.17_transcript.html\"> transcript for \u201cCP 4.17\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<h2>Partial Differential Equations<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A <strong>partial differential equation<\/strong> (PDE) involves an unknown function of multiple variables and its partial derivatives. These equations model physical phenomena like heat flow, wave propagation, and fluid dynamics.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Common Examples<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Heat equation<\/strong>: [latex]u_t = c^2(u_{xx} + u_{yy})[\/latex] describes temperature distribution over time<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Wave equation<\/strong>: [latex]u_{tt} = c^2(u_{xx} + u_{yy})[\/latex] describes vibrations and wave motion<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Laplace&#8217;s equation<\/strong>: [latex]u_{xx} + u_{yy} = 0[\/latex] describes steady-state phenomena<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">In these equations, [latex]x[\/latex] and [latex]y[\/latex] represent spatial coordinates, and [latex]t[\/latex] represents time.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Verifying Solutions<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">To verify a proposed solution:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Calculate all required partial derivatives<\/li>\n<li class=\"whitespace-normal break-words\">Substitute them into the PDE<\/li>\n<li class=\"whitespace-normal break-words\">Simplify to check if both sides match<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>Visualizing Solutions<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Solutions to PDEs are functions of multiple variables. For the heat equation with time, we can graph snapshots at different times to see how temperature evolves. As time increases, extreme temperatures level out.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Calculate the needed partial derivatives carefully. Substitute into the PDE. Simplify both sides. Verify equality.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Verify that [latex]u\\,(x,\\ y,\\ t)=2\\,\\sin\\,(\\frac{x}{3})\\,\\sin\\,(\\frac{y}{4})\\,e^{-25t\/16}[\/latex] is a solution to the heat equation<\/p>\n<p style=\"text-align: center;\">[latex]u_{t}=9(u_{xx}+u_{yy})[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1367573930124\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1367573930124\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\"><\/div>\n<p>[latex]u\\,(x,\\ y,\\ t)=2\\,\\sin\\,(\\frac{x}{3})\\,\\sin\\,(\\frac{y}{4})\\,e^{-25t\/16}[\/latex] is a solution to the heat equation. See worked example video below to verify the solution.\n<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186156&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=pMoH9Key-B8&amp;video_target=tpm-plugin-k0fmirvu-pMoH9Key-B8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.18_transcript.html\">transcript for \u201cCP 4.18\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-601","chapter","type-chapter","status-publish","hentry"],"part":426,"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\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/601","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/601\/revisions"}],"predecessor-version":[{"id":1835,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/601\/revisions\/1835"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/426"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/601\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=601"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=601"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=601"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=601"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}