Calculus I:
functions of several variables
One variable function
The definition of the derivative
Differential

One variable function
Differential

We know the derivation of the functions of a single variable. Here we are going to see how to extend this notion to the case of the functions of several variables.

Most of the statements in this chapter will only concern the functions of two variables, the case of the functions of three variables or more is easily deduced.

Since we are going to generalize the notion of a derivative to the functions of two variables, let us first recall a few definitions and notations for the functions of a single variable.

Definition

f: D → R
x → f (x)

We say that f is differentiable in x, and f'(x) its derivative when the next limit is finite, that is the limit exists and it is not +∞ or - ∞.



Notation :

Another way of writing f'(x) is the following :



This notation recalls that f'(x) is the quotient of a "small difference" on f and of a "small difference" on x, because h = (x + h) - x.

We then obtain:

df = f'(x) dx.