# What are partial derivatives?

Feb 19, 2015

Hello,

If $f$ is a function with 2 variables, for example

$f \left(x , y\right) = {x}^{2} y + \setminus \cos \left(x y\right)$

you can calculate 2 derivatives :

1) the derivative on $x$ (then $y$ is like a constant) : $f {'}_{x} \left(x , y\right)$ or $\setminus \frac{\setminus \partial f}{\setminus \partial x} \left(x , y\right)$.

2) the derivative on $y$ (then $x$ is like a constant) : $f {'}_{y} \left(x , y\right)$ or $\setminus \frac{\setminus \partial f}{\setminus \partial y} \left(x , y\right)$.

For example, with $f \left(x , y\right) = {x}^{2} y + \setminus \cos \left(x y\right)$, you have

$\setminus \frac{\setminus \partial f}{\setminus \partial x} \left(x , y\right) = 2 x y - y \setminus \sin \left(x y\right)$ and

$\setminus \frac{\setminus \partial f}{\setminus \partial y} \left(x , y\right) = {x}^{2} - x \setminus \sin \left(x y\right)$.