Partial derivative - Knowlpedia

Let $U\subseteq\mathbb R^n$ be open, $f:U\to\mathbb R^m$ , and $a\in U$ . The partial derivative of $f$ at $a$ with respect to the $j$ -th coordinate is

\frac{\partial f}{\partial x_j}(a) =\lim_{t\to0}\frac{f(a+t e_j)-f(a)}{t},

when this limit exists in $\mathbb R^m$ .

Partial derivatives are the entries of the Jacobian matrix. Existence of every partial derivative at $a$ does not by itself imply differentiability at $a$ .

Examples

For $f(x,y)=x^2y$ , one has $\frac{\partial f}{\partial x}=2xy$ and $\frac{\partial f}{\partial y}=x^2$ .
For $f(x,y)=|x|$ , the partial derivative $\frac{\partial f}{\partial x}(0,y)$ does not exist.