mixed partial derivative

A second-order partial derivative taken with respect to two different variables, such as $f_{xy}$ .

$f_{xy}$ means "differentiate with respect to $x$ first, then $y$ " — written $(f_x)_y$ or $\dfrac{\partial^2 f}{\partial y\,\partial x}$ — while $f_{yx}$ differentiates in the opposite order. These are the two mixed partial derivatives of $f$ , as opposed to $f_{xx}$ and $f_{yy}$ , which differentiate twice with respect to the same variable.

Examples

For $f(x,y)=x^3y^2-xy$ , $f_{xy}=(f_x)_y=6x^2y-1$ and $f_{yx}=(f_y)_x=6x^2y-1$ .