# Derivation.

Definition 3.2.1   Let be a function defined on a domain in and let be an interior point of . The function is derivable (or differentiable) at if there exists a complex number such that

We denote this by .

Remark 3.2.2   Another formulation of Def. 2.1 is as follows:

Example 3.2.3   Let , where . Then, for every :

from what follows:

The function is differentiable at every point and .

The algebra of derivable functions of a complex variable is (formally) identical to the algebra of derivable functions of a real variable seen in Calculus.

Proposition 3.2.4   Let and be two functions defined on a neighborhood of . We suppose that and are continuous at .
(i)
is differentiable at and .
(ii)
is differentiable at and we have:

(iii)
If , then is derivable at and we have .
(iv)
If ,then is derivable at and we have:

Example 3.2.5   If , then is differentiable at every point of and .

Proposition 3.2.6   If is derivable at and if is derivable at , then is derivable at and we have:

As easy consequences of Proposition 2.4, we have Corollary 2.7 and Corollary 2.8.

Corollary 3.2.7        A polynomial function is differentiable on the whole of .

Corollary 3.2.8        A rational function is differentiable at every point of its domain of definition

Example 3.2.9   The function such that is differentiable on and we have:

Actually, we do not dispose of enough functions in order to display more interseting examples. For that purpose, we shall wait until next chapter (i.e. Chapter chapter analytic functions).

Noah Dana-Picard 2007-12-24