Partial derivatives

Summary

Functions of two variables

$f$ is a function of two variables and $\left( x,y \right)$ is in the domain of $f$ .

$f'_x\left( x,y \right)=\lim_{h→0} \frac{f\left( x+h,y \right)-f\left( x,y \right)}{h}$

is called the partial derivative of $f$ with respect to $x$ at $\left( x,y \right)$ (assuming that the limit exists)
Similarly,

$f'_y\left( x,y \right)=\lim_{h→0} \frac{f\left( x,y+h \right)-f\left( x,y \right)}{h}$

is called the partial derivative of $f$ with respect to $y$ at $\left( x,y \right)$ (assuming that the limit exists)
If both partial derivatives $f'_x\left( x,y \right)$ and $f'_y\left( x,y \right)$ exists and are continuous close to $(x,y)$ then $f$ is differentiable at $(x,y)$ . (This is not the formal definition of differentiability but a sufficient condition that is easier to check).

Functions of $n$ variables

If $f$ is a function of $n$ variables and if $x_0=\left( x_1,⋯,x_n \right)$ is in the domain of $f$ and the limit

$\lim_{h→0} \frac{f\left( x_1,⋯,x_i+h,⋯,x_n \right)-f\left( x_1,⋯,x_n \right)}{h}$

exists then the limit is called the partial derivative of $f$ with respect to $x_i$ at $x$ denoted by $f'_i(x)$ .
If $f$ is a function of $n$ variables then the $n×1$ column vector

$\begin{pmatrix} f'_1(x) \\ f'_2(x) \\ \vdots \\ f'_n(x) \end{pmatrix}$