The function f(x,y) is a function of the two variables x and y. Suppose f(x,y) is continuous and it has second derivatives in all directions. Recall that fx(x,y) and fy(x,y) are the partial derivatives of f(x,y) with respect to x and y, respectively. We may define second order partial derivatives in the following manner:

We will write

If f(x,y) is continuously twice-differentiable in all directions, then the mixed partial derivatives, fxy(x,y) and fyx(x,y) are equal. For example, if

f(x,y) = x3 + y3 - 6xy

then