Mixed partial derivative

A second partial derivative taken with respect to two different coordinates

Mixed partial derivative

A mixed partial derivative of a scalar function $f:U\to \mathbb{R}$ (with $U\subseteq \mathbb{R}^n$ ) at $a\in U$ is a second-order partial derivative of the form

\frac{\partial^2 f}{\partial x_i\,\partial x_j}(a) =\frac{\partial}{\partial x_i}\left(\frac{\partial f}{\partial x_j}\right)(a),

provided the relevant partial derivatives exist.

Mixed partial derivatives form the off-diagonal entries of the Hessian matrix . Under appropriate regularity hypotheses (for example, continuity of the second partials near $a$ ), the Schwarz–Clairaut theorem guarantees equality of the two orders of differentiation.

Examples:

For $f(x,y)=x^2y$ , one has $\frac{\partial^2 f}{\partial x\,\partial y}(x,y)=2x$ and $\frac{\partial^2 f}{\partial y\,\partial x}(x,y)=2x$ .
For $f(x,y)=e^{x+y}$ , every mixed partial derivative exists and equals $e^{x+y}$ .