skip to main content

Mathematics (thesaurus)

Content language

Home / Explore / Browse / Vocabularies / Mathematics (thesaurus)

... > function > elementary function > algebraic function > reciprocal function > inverse function rule

... > real analysis > real-valued function > algebraic function > reciprocal function > inverse function rule

mathematical analysis > calculus > differential calculus > inverse function rule

Preferred term

inverse function rule

Definition(s)

In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function $f$ in terms of the derivative of $f$ . More precisely, if the inverse of ${\\displaystyle f}$ is denoted as ${\\displaystyle f^{-1}}$ , where ${\\displaystyle f^{-1}(y)=x}$ if and only if ${\\displaystyle f(x)=y}$ , then the inverse function rule is, in Lagrange's notation,

${\\displaystyle \\left[f^{-1}\ ight]'(a)={\ rac {1}{f'\\left(f^{-1}(a)\ ight)}}}$ .

This formula holds in general whenever ${\\displaystyle f}$ is continuous and injective on an interval $I$ , with ${\\displaystyle f}$ being differentiable at ${\\displaystyle f^{-1}(a)}$ ( ${\\displaystyle \\in I}$ ) and where ${\\displaystyle f'(f^{-1}(a))\ eq 0}$ . The same formula is also equivalent to the expression

${\\displaystyle {\\mathcal {D}}\\left[f^{-1}\ ight]={\ rac {1}{({\\mathcal {D}}f)\\circ \\left(f^{-1}\ ight)}},}$

where ${\\displaystyle {\\mathcal {D}}}$ denotes the unary derivative operator (on the space of functions) and ${\\displaystyle \\circ }$ denotes function composition.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Inverse_function_rule)

Broader concept(s)

In other languages

règle de dérivation des fonctions réciproques

French

URI

http://data.loterre.fr/ark:/67375/PSR-GXT3JJHT-8

Download this concept:

RDF/XML TURTLE JSON-LD Created 8/2/23, last modified 8/2/23