Partial derivative

A partial derivative of a map $f:U\to \mathbb{R}^m$ (with $U\subseteq \mathbb{R}^n$) at $a=(a_1,\dots,a_n)\in U$ with respect to the $j$th coordinate is the limit

when it exists (for vector-valued $f$, this limit is taken in $\mathbb{R}^m$).

Partial derivatives are one-coordinate versions of the limit at a point and are the entries used to build the Jacobian matrix . Existence of all partial derivatives at $a$ does not by itself guarantee that $f$ is a differentiable map at $a$.

Examples:

• For $f(x,y)=x^2y$, one has $\frac{\partial f}{\partial x}(x,y)=2xy$ and $\frac{\partial f}{\partial y}(x,y)=x^2$.
• For $f(x,y)=|x|$, the partial derivative $\frac{\partial f}{\partial x}(0,y)$ does not exist (for any $y$), since the corresponding one-variable derivative at $0$ fails to exist.