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mathematical analysis > calculus > differential calculus > inverse function rule

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inverse function rule

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)

règle de dérivation des fonctions réciproques

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