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 f_{x}(x,y) and
f_{y}(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

- ``f
_{xx}(x,y)'' to represent**the partial derivative of f**,_{x}(x,y) with respect to x - ``f
_{xy}(x,y)'' to represent**the partial derivative of f**,_{x}(x,y) with respect to y - ``f
_{yx}(x,y)'' to represent**the partial derivative of f**, and_{y}(x,y) with respect to x - ``f
_{yy}(x,y)'' to represent**the partial derivative of f**._{y}(x,y) with respect to y

f(x,y) = x^{3} + y^{3} - 6xy

then

- f
_{x}(x,y) = 3x^{2}- 6y, - f
_{y}(x,y) = 3y^{2}- 6x, - f
_{xx}(x,y) = 6x, - f
_{xy}(x,y) = -6, - f
_{yx}(x,y) = -6, and - f
_{yy}(x,y) = 6y.