The function f(x,y) is a function of the two variables x and y. If f(x,y) is continuous and it has derivatives in all directions, then we may define partial derivatives in the following manner:

We will write ``fx(x,y)'' to represent the partial derivative of f(x,y) with respect to x. Symbolically, this is computed by computing the derivative of f with respect to x, treating the occurrences of the variable y as constants. For example, if

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

then fx(x,y) = 3x2 + 6y.

Likewise, fy(x,y) is the partial derivative with respect to y, symbolically computed by taking the derivative with respect to y, treating x as a constant:

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

fy(x,y) = 3y2 + 6x.