Math 22 Partial Derivatives

Partial Derivatives of a Function of Two Variables

 If $z=f(x,y)$, then the first partial derivatives of  with respect to $x$ and $y$ are the functions Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\partial z}{\partial x}}}
 and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\partial z}{\partial x}}}
, defined as shown.
 
 ${\frac {\partial z}{\partial x}}=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x,y)-f(x,y)}{\Delta x}}$
 
 ${\frac {\partial z}{\partial y}}=\lim _{\Delta y\to 0}{\frac {f(x,y+\Delta y)-f(x,y)}{\Delta y}}$
 
 We can denote Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\partial z}{\partial x}}}
 as $f_{x}(x,y)$ and ${\frac {\partial z}{\partial y}}$ as $f_{y}(x,y)$

Example: Find Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\partial z}{\partial x}}} and ${\frac {\partial z}{\partial y}}$ of:

1) $z=f(x,y)=2x^{2}-4xy$

Solution:
${\frac {\partial z}{\partial x}}=4x^{2}-4y$
${\frac {\partial z}{\partial y}}=-4x$

2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=f(x,y)=x^2y^3}

Solution:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial z}{\partial x}=2xy^3}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial z}{\partial y}=3x^2y^2}

3) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=f(x,y)=x^2e^{x^2y}}

Solution:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial z}{\partial x}=2xe^{x^2y}+x^2e^{x^2y}2xy} (product rule +chain rule)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial z}{\partial y}=x^2e^{x^2y}(x^2)=x^4e^{x^2y}}

Higher-Order Partial Derivatives

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial}{\partial x}(\frac{\partial f}{\partial x})=\frac{\partial^2 f}{\partial x^2}=\f_{xx}} Return to Topics Page

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