Operations on the derivative

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 $f$ and $g$ two functions that one supposes derivable.

Linearity

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: $\backslash \; bigl\; (\backslash \; alpha\; F\; \backslash \; bigr)\; \text{'}=\; \backslash \; alpha\; f\text{'}$, $\backslash \; bigl\; (f+g\; \backslash \; bigr)\; \text{'}=\; f\text{'}+g\text{'}$.

Composition

The composition of two derivable functions is derivable, where it is defined (precisely on the reciprocal image by $f$ of the field of definition of $g$) and is calculated according to the rule: $(G\; \backslash \; circ\; F)\; “=\; F”\; \backslash \; cdot\; (g\text{'}\; \backslash \; circ\; F)$. This rule admits for consequences the rule of derivation of reciprocal of a bijection: $\backslash \; bigl\; (f^\; \{-\; 1\}\; \backslash \; bigr)\; “=\; \backslash \; frac\; \{1\}\; \{F”\; \backslash \; circ\; f^\; \{-\; 1\}\},$ and calculation of the reverse of a function regulates it (one places oneself on an interval on whom $f$ is not cancelled), by using the elementary calculation of derived from the function $x\; \backslash \; mapsto\; \backslash \; tfrac1x$: $\backslash \; left\; (\backslash \; frac\; \{1\}\; \{F\}\; \backslash \; right)\; “=\; -\; \backslash \; frac\; \{F”\}\; \{f^2\}$

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: $\backslash \; bigl\; (fg\; \backslash \; bigr)\; \text{'}=\; f\text{'}\; G\; +\; fg\text{'}$.

In particular, this relation admits like consequences the rule of derivation of the powers:

$\backslash \; bigl\; (f^n\; \backslash \; bigr)\; \text{'}=\; nf\text{'}\; f^\; \{n-1\}$, and that for the derivative of a quotient: $\backslash \; left\; (\backslash \; frac\; \{F\}\; \{G\}\; \backslash \; right)\; “=\; \backslash \; frac\; \{f\text{'}\; g-fg”\}\; \{g^2\}$

Random links:

Fleury-in-beer | Large-Charmont | Canton of Chartres-South-West | Fatal beauty (film, 1987) | Coumans | Le_Kansas,_l'Illinois

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org