Mixed partial derivative ($f_{xy}$)

A mixed partial derivative takes partial derivatives in different variables, for example $f_{xy}={\partial }_y({\partial }_{x}f)$. If both second partial derivatives are continuous in a neighbourhood, their order can be interchanged: $f_{xy}=f_{yx}$. The chain rule transforms mixed partial derivatives under a linear change of variables by composing the corresponding first-order differential operators.