Let *f* be a function of two variables *x*,*y*, defined on a neighborhood of the point
*P*_{0}=(*x*_{0},*y*_{0}).

- The
*first partial derivative*of*f*at*P*_{0}with respect to the variable*x*is

provided the limit exists and is finite. - The
*first partial derivative*of*f*at*P*_{0}with respect to the variable*y*is

provided the limit exists and is finite.

For a function of more than two variables, we define the other partail derivatives by the same way.

In each case, we consider one variable as a constant and differentiate with respect to the other variable, according to the well-known rules of differentiation, as learnt in Calculus I.

We differentiate both sides of the relation with respect to *x*:

__Notations:__
and
.