Math 22 Partial Derivatives

Partial Derivatives of a Function of Two Variables

 If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle z=f(x,y)}
, then the first partial derivatives of  with respect to $x$ and 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 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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) $z=f(x,y)=x^{2}y^{3}$

Solution:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\partial z}{\partial x}}=2xy^{3}}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\partial z}{\partial y}}=3x^{2}y^{2}}

3) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle z=f(x,y)=x^{2}e^{x^{2}y}}

Solution:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\partial z}{\partial x}}=2xe^{x^{2}y}+x^{2}e^{x^{2}y}2xy} (product rule +chain rule)
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\partial z}{\partial y}}=x^{2}e^{x^{2}y}(x^{2})=x^{4}e^{x^{2}y}}

Higher-Order Partial Derivatives

1. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\partial }{\partial x}}({\frac {\partial f}{\partial x}})={\frac {\partial ^{2}f}{\partial x^{2}}}=f_{xx}}

2. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\partial }{\partial y}}({\frac {\partial f}{\partial y}})={\frac {\partial ^{2}f}{\partial y^{2}}}=f_{yy}}

3. ${\frac {\partial }{\partial y}}({\frac {\partial f}{\partial x}})={\frac {\partial ^{2}f}{\partial y\partial x}}=f_{xy}$

4. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\partial }{\partial x}}({\frac {\partial f}{\partial y}})={\frac {\partial ^{2}f}{\partial x\partial y}}=f_{yx}}

1) Find Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f_{xy}} , given that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x,y)=2x^{2}-4xy} ,

Solution:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f_{x}=4x-4y}
Then, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f_{xy}=-4}

2) Find Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f_{yx}} , given that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle z=f(x,y)=3xy^{2}-2y+5x^{2}y^{2}} ,

Solution:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f_{y}=6xy-2+10x^{2}y}
Then, $f_{yx}=6y+20xy$

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