{"id":218,"date":"2019-09-10T16:20:27","date_gmt":"2019-09-10T16:20:27","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=218"},"modified":"2020-06-18T17:39:05","modified_gmt":"2020-06-18T17:39:05","slug":"partial-differentiation","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/multivariable-calculus\/partial-differentiation\/","title":{"rendered":"Partial Differentiation"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Partial Differentiation &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\nSuppose you want to forecast the weather this weekend in Los Angeles.\nYou construct a formula for the temperature as a function of several\nenvironmental variables, each of which is not entirely predictable.\nNow you would like to see how your weather forecast would change as\none particular environmental factor changes, holding all the other\nfactors constant.  To do this investigation, you would use the concept\nof a <b>partial derivative<\/b>.\n\n<\/p>\n\n\n\n<p> Let the temperature $T$ depend on variables $x$ and $y$, $T = f(x,y)$.  The rate of change of $f$ with respect to $x$, holding $y$ constant, is called the <b>partial derivative of $f$ with respect to $x$<\/b> and is denoted by $f_{x}(x,y)$.  Similarly, the rate of change of $f$ with respect to $y$ is called the <b>partial derivative of $f$ with respect to $y$<\/b>  and is denoted by $f_{y}(x,y)$. <\/p>\n\n\n\n<p>\nWe define \n\n<\/p>\n\n\n\n<center>\n<p align=\"center\">\n\t\t$$\n\t\tf_{x}(x,y) = \\lim_{h \\rightarrow 0} \\frac{f(x+h,y)-f(x,y)}{h} \\phantom{.}\n\t\t$$\n\t\t$$\n\t\tf_{y}(x,y) = \\lim_{h \\rightarrow 0} \\frac{f(x,y+h)-f(x,y)}{h}.\n\t\t$$\n<\/p>\n\n<p>\n<b>Do you see the similarity beween these and the limit <br> definition of a function of one variable?<\/b>\n<\/p><\/center>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\n\t$\\qquad\\qquad \\begin{array}{rcl@{\\qquad}rcl}\n\t\t\\displaystyle\n\t\t\\mathrm{Let~}f(x,y) &amp; = &amp; xy^{2} &amp; &amp; &amp; \\\\\n\t\t\\mathrm{Then~}f_{x}(x,y) &amp; = &amp; \\lim\\limits_{h \\rightarrow 0} \n\t\t\\frac{(x+h)y^{2}-xy^2}{h} &amp; f_{y}(x,y) &amp; = &amp; \\lim\\limits_{h \\rightarrow 0} \n\t\t\\frac{x(y+h)^{2}-xy^2}{h}\\\\\n\t\t&amp; = &amp; \\lim\\limits_{h \\rightarrow 0} \\frac{hy^{2}}{h} &amp; &amp; = &amp; \\lim\\limits_{h\n\t\t\\rightarrow 0} \\frac{2xyh+ xh^{2}}{h} \\\\\n\t\t&amp; = &amp; y^{2}. &amp; &amp; = &amp; \\lim\\limits_{h \\rightarrow 0} (2xy + xh) \\\\\n\t\t&amp; &amp; &amp; &amp; = &amp; 2xy.\n\t\\end{array}$\n\n<\/p>\n\n\n\n<p>\nIn practice, we use our knowledge of single-variable calculus to\ncompute partial derivatives.  To calculate $f_{x}(x,y)$, you view $y$\nas a constant and differentiate $f(x,y)$ with respect to $x$:\n$$\nf_{x}(x,y) =y^{2} \\mathrm{~as~expected~since~} \\frac{d}{dx}[x]= 1.\n$$\nSimilarly,\n$$\nf_{y}(x,y) = 2xy \\mathrm{~since~} \\frac{d}{dy}\\left[y^{2}\\right]= 2y.\n$$\n\n<\/p>\n\n\n<style>.kt-accordion-id_6100bb-15 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_6100bb-15 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_6100bb-15 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_6100bb-15 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_6100bb-15 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_6100bb-15 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_6100bb-15 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6100bb-15:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_6100bb-15 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_6100bb-15 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_6100bb-15 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_673179-1b\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">More Examples<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>\n<!------------------------>\n\nLet $f(x,y)= 3x^{2}+ e^{-xy^{2}}$,\n\n<\/p>\n\n\n\n<p>\nThen\n$$\nf_{x}(x,y) = 6x -y^{2}e^{-xy^{2}} \\quad \\left( \\mathrm{since~}\n\\frac{d}{dx}\\left[ 3x^{2}\\right] = 6x \\mathrm{~and~}\n\\frac{d}{dx}\\left[ e^{-cx}\\right] = -ce^{-cx} \\right)\n$$\n$$\nf_{y}(x,y) = -2xye^{-xy^{2}} \\quad \\left( \\mathrm{since~}\n\\frac{d}{dy}\\left[ c \\right] = 0 \\mathrm{~and~}\\frac{d}{dy}\\left\n[ e^{-cy^{2}}\\right] = -2cye^{-cy^{2}}\\right).\n$$\n\n<\/p>\n\n\n\n<p>\nLet $f(x,y)= y\\cos (xy)$,\n\n<\/p>\n\n\n\n<p>\nThen\n\n<\/p>\n\n\n\n<p>\n$\\quad \\displaystyle\nf_{x}(x,y) = -y^{2}\\sin (xy)\n$\n\n<\/p>\n\n\n\n<p>\n$\\quad \\displaystyle\nf_{y}(x,y) = \\cos (xy) -xy\\sin (xy) \\mathrm{~using~the~product~rule}.\n$\n\n\n\n<!------------------------>\n\n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Notation<\/h4>\n\n\n\n<ul class=\"wp-block-list\"><li> Let $z = f(x,y)$.\n\n\t\t<p>\n\t\tThe partial derivative $f_{x}(x,y)$ can also be written as\n\t\t$$\n\t\t\\frac{\\partial f}{\\partial x}(x,y) \\mathrm{~or~} \\frac{\\partial\n\t\tz}{\\partial x}.\n\t\t$$\n\t\tSimilarly, $f_{y}(x,y)$ can also be written as\n\t\t$$\n\t\t\\frac{\\partial f}{\\partial y}(x,y) \\mathrm{~or~} \\frac{\\partial\n\t\tz}{\\partial y}.\n\t\t$$\n\n\t<br><br>\n\t<\/p><\/li><li> The partial derivative $f_{y}(x,y)$ evaluated at the point\n\t\t$(x_{0},y_{0})$ can be expressed in several ways:\n\t\t$$\n\t\tf_{x}(x_{0},y_{0}) \\mathrm{,~} \\left. \\frac{\\partial f}{\\partial x}\n\t\t\\right|_{(x_{0},y_{0})} \\mathrm{,~or~} \\frac{\\partial f}{\\partial\n\t\tx}(x_{0},y_{0}) .\n\t\t$$\n\n\t\t<p>\n\t\tThere are analogous expressions for $f_{y}(x_{0},y_{0})$.\n\n<\/p><\/li><\/ul>\n\n\n\n<p><\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Geometrical Meaning<\/h4>\n\n\n\n<ul class=\"wp-block-gallery alignright columns-1 is-cropped wp-block-gallery-1 is-layout-flex wp-block-gallery-is-layout-flex\"><li class=\"blocks-gallery-item\"><figure><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"225\" height=\"300\" src=\"https:\/\/i0.wp.com\/104.42.120.246.xip.io\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/partial.gif?resize=225%2C300\" alt=\"Follow the image link for a complete description of the image\" data-id=\"378\" class=\"wp-image-378\"\/><\/figure><\/li><\/ul>\n\n\n\n<p>\n\n\nSuppose the graph of $z = f(x,y)$ is the surface shown.  Consider the\npartial derivative of $f$ with respect to $x$ at a point\n$(x_{0},y_{0})$. \n\n<\/p>\n\n\n\n<p>\nHolding $y$ constant and varying $x$, we trace out a curve that is the\nintersection of the surface with the vertical plane $y= y_{0}$. \n\n<\/p>\n\n\n\n<p> The partial derivative $f_{x}(x_{0},y_{0})$ measures the change in $z$ per unit increase in $x$ along this curve.  That is, $f_{x}(x_{0},y_{0})$ is just the slope of the curve at $(x_{0},y_{0})$.  The geometrical interpretation of $f_{y}(x_{0},y_{0})$ is analogous.  <\/p>\n\n\n\n<p><\/p>\n\n\n\n<center>\n<h4>Notes<\/h4>\n<\/center>\n\n\n\n<ul class=\"wp-block-list\"><li> <b>Functions of More than Two Variables<\/b> <p> For $g(x,y,z)$, the partial derivative $g_{x}(x,y,z)$ is calculated by holding $y$ and $z$ constant and differentiating with respect to $x$. The partial derivatives $g_{y}(x,y,z)$ and $g_{z}(x,y,z)$ are calculated in an analagous manner.<\/p><\/li><\/ul>\n\n\n<style>.kt-accordion-id_588f00-06 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_588f00-06 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_588f00-06 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_588f00-06 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_588f00-06 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_588f00-06 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_588f00-06 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_588f00-06:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_588f00-06 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_588f00-06 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_588f00-06 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_7f2d1d-44\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Example<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>\n<!------------------------>\n\nLet $f(x,y,z) = x^{2}z-3xy^{4}$,\n\n<\/p>\n\n\n\n<p>\nThen\n$$\nf_{x}(x,y,z) = 2xz-3y^{4}\n$$\n$$\nf_{y}(x,y,z) = -12xy^{3}\n$$\n$$\nf_{z}(x,y,z) = x^{2}.\n$$\n\n\n\n<!------------------------>\n\n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> <strong>Higher-Order Partial Derivatives<\/strong> <br><br> For a function $f(x,y)$, the partial derivatives $\\frac{\\partial f}{\\partial x}$ and $\\frac{\\partial f}{\\partial y}$ are themselves functions of $x$ and $y$, so we can take partial derivatives of them:   <br><br>$$ f_{xx}=  \\frac{\\partial}{\\partial x} \\left( \\frac{\\partial f}{\\partial x} \\right) = \\frac{\\partial^{2} f}{\\partial x^{2}} \\qquad f_{xy} =  \\frac{\\partial}{\\partial y} \\left( \\frac{\\partial f}{\\partial x} \\right) = \\frac{\\partial^{2} f}{\\partial y \\partial x} \\phantom{.} $$  $$ f_{yy} = \\frac{\\partial}{\\partial y} \\left( \\frac{\\partial f}{\\partial y} \\right) = \\frac{\\partial^{2} f}{\\partial y^{2}} \\qquad  f_{yx} = \\frac{\\partial}{\\partial x} \\left( \\frac{\\partial f}{\\partial y} \\right) = \\frac{\\partial^{2} f}{\\partial x \\partial y}. $$ Higher-order partial derivatives (e.g. $f_{xxy}$) can also be calculated.  Using the subscript notation, the order of differentiation is from left to right. <br><br>$f_{xy}$ and $f_{yx}$ are called mixed second-order partial derivatives. If $f$, $f_{x}$, $f_{y}$, $f_{xy}$, and $f_{yx}$ are continuous on an open region, then $f_{xy} = f_{yx}$ at each point in the region, so the order in which the differentiation is done does not matter. <\/li><\/ul>\n\n\n<style>.kt-accordion-id_21bd1e-cd .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_21bd1e-cd .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_21bd1e-cd > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_21bd1e-cd > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_21bd1e-cd .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_21bd1e-cd .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_21bd1e-cd > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_21bd1e-cd:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_21bd1e-cd .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_21bd1e-cd .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_21bd1e-cd kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_ae7fea-aa\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Example<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>\n<!------------------------>\n\nLet $f(x,y) = x^{2}-4xy^{3}$,\n\n<\/p>\n\n\n\n<p>\nThen\n$$\nf_{x}(x,y) = 2x &#8211; 4y^{3}\n$$\n$$\nf_{y}(x,y) = -12xy^{2}\n$$\n$$\nf_{xy}(x,y) = f_{yx}(x,y) = -12y^{2}\n$$\n$$\nf_{xx}(x,y) = 2\n$$\n$$\nf_{yy}(x,y) = -24xy.\n$$\n\n<\/p>\n\n\n\n<p>\nHigher-order partial derivatives can be computed as well!\n\n\n\n<!------------------------>\n\n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<p>\n<\/p><h4>Key Concepts<\/h4>\n<p>\n<\/p><\/center>\n\n\n\n<p>\n\nConsider a function $f(x,y)$.\n<\/p>\n\n\n\n<center>\n\t<table align=center border=0>\n\t\t<tr>\n\t\t\t<td> $$ f_{x}(x,y) $$\n\t\t\t<td> $\\quad = \\quad$\n\t\t\t<td> rate of change of $f$ <BR> with respect to $x$\n\t\t\t<td> $\\quad = \\quad$\n\t\t\t<td> $$ \\lim_{h \\rightarrow 0} \\frac{f(x+h, y)- f(x,y)}{h} \\phantom{.} $$\n\t\t<tr>\n\t\t\t<td> $$ f_{y}(x,y) $$ \n\t\t\t<td> $\\quad = \\quad$\n\t\t\t<td> rate of change of $f$ <BR> with respect to $y$\n\t\t\t<td> $\\quad = \\quad$\n\t\t\t<td> $$ \\lim_{h \\rightarrow 0} \\frac{f(x, y+h)- f(x,y)}{h}. $$\n\t<\/table>\n<\/center>\n\n\n\n<p>\nTo calculate $f_{x}(x,y)$, differentiate $f$ with respect to $x$\nholding $y$ constant.  Similarly, to calculate $f_{y}(x,y)$,\ndifferentiate $f$ with respect to $y$ holding $x$ constant.\n\n<!------------------------>\n\n\n<br>\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p> [<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ2910\/\">I&#8217;m ready to take the quiz.<\/a>] [<a href=\"#top\">I need to review more.<\/a>]<br> <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Partial Differentiation &#8211; HMC Calculus Tutorial Suppose you want to forecast the weather this weekend in Los Angeles. You construct a formula for the temperature as a function of several environmental variables, each of which is not entirely predictable. Now you would like to see how your weather forecast would change as one particular environmental&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":59,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-218","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/218","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=218"}],"version-history":[{"count":10,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/218\/revisions"}],"predecessor-version":[{"id":1241,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/218\/revisions\/1241"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/59"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=218"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=218"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}