# 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.
${\displaystyle {\frac {\partial z}{\partial x}}=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x,y)-f(x,y)}{\Delta x}}}$
${\displaystyle {\frac {\partial z}{\partial y}}=\lim _{\Delta y\to 0}{\frac {f(x,y+\Delta y)-f(x,y)}{\Delta y}}}$