Second Partial Derivatives

Given a function z = f(x, y) of two variables, the partial derivatives f_x(x, y) and f_y(x, y) may themselves be differentiated with respect to either x or y. Thus there are four possible second partial derivatives. Here they are.

Twice with respect to x:

f_xx, or

Twice with respect to y:

f_yy, or

First with respect to x and then with respect to y:

(f_x)_y = f_xy, or

First with respect to y and then with respect to x:

(f_y)_x = f_yx, or

Similar notation is used for three or more variables and for higher partial derivatives.