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 ${\frac {\partial z}{\partial x}}$ and ${\frac {\partial z}{\partial x}}$, defined as shown.
 
 ${\frac {\partial z}{\partial x}}=\lim _{\delta x\to 0}{\frac {f(x+\delta x,y)-f(x,y)}{\delta x}}$
 
 ${\frac {\partial z}{\partial y}}=\lim _{\delta y\to 0}{\frac {f(x,y+\delta y)-f(x,y)}{\delta y}}$

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