The calculation of the Dérivée from some functions with real values or complex (or more generally in a topological body) can be carried out by using a certain number of operations on the derivative , in particular some related to the operations on the real numbers and complexes. The demonstrations of these properties derive from the Opérations on the limits.
In all the article, one notes and two functions that one supposes derivable.
Derivation is a linear Opérateur, i.e. the space of the derivable functions is stable by nap and multiplication of its elements by realities (it is a real vector space), and the following relations are checked:
,
.
Composition
The composition of two derivable functions is derivable, where it is defined (precisely on the reciprocal image by
of the field of definition of
) and is calculated according to the rule:
.
This rule admits for consequences the rule of derivation of reciprocal of a bijection:
and calculation of the reverse of a function regulates it (one places oneself on an interval on whom
is not cancelled), by using the elementary calculation of derived from the function
:
Product and quotient
Derivation is a differential Opérateur, i.e. the space of the derivable functions is stable by multiplication, and the Formule of Leibniz is checked:
.
In particular, this relation admits like consequences the rule of derivation of the powers:
,
and that for the derivative of a quotient: