Definition

Here is a precise definition.

When f_x(a, b) exists, it is equal to the standard part

for any nonzero infinitesimal Δx. Similarly when f_y(a, b) exists,

for any nonzero infinitesimal Δy.

Just as the one-variable derivative f'(x) is a function of x, the partial derivatives f_x(x, y) and f_y(x, y) are again functions of x and y. At each point (x, y), the partial derivative f_x(x, y) either has exactly one value or is undefined.