# Difference between revisions of "Math 22 Partial Derivatives"

 If ${\displaystyle z=f(x,y)}$, then the first partial derivatives of  with respect to ${\displaystyle x}$ and ${\displaystyle y}$ are the functions ${\displaystyle {\frac {\partial z}{\partial x}}}$ and ${\displaystyle {\frac {\partial z}{\partial x}}}$, defined as shown.
${\displaystyle {\frac {\partial z}{\partial x}}=\lim _{\delta x\to 0}{\frac {f(x+\delta x,y)-f(x,y)}{\delta x}}}$
${\displaystyle {\frac {\partial z}{\partial y}}=\lim _{\delta y\to 0}{\frac {f(x,y+\delta y)-f(x,y)}{\delta y}}}$