File:Umkehrregel 2.png

w:en:Inverse functions and differentiation, w:de:Umkehrregel

Shown functions:

${\displaystyle {\color {Blue}f(x)=\log _{2}x}~~;~~~~{\color {Red}f^{-1}(x)=2^{x}}}$

${\displaystyle {\color {Periwinkle}f'(x)={\frac {1}{\ln 2\cdot x}}}~~;~~~~{\color {Salmon}(f^{-1})'(x)=\ln 2\cdot 2^{x}}}$

Rule:

${\displaystyle {\color {Periwinkle}f'}(x)={\frac {1}{{\color {Salmon}(f^{-1})'}({\color {Blue}f}(x))}}}$

The illustration shows the validity of the rule for the random ${\displaystyle x_{0}={\frac {4}{\ln 2}}\approx 5.77}$:

${\displaystyle {\color {Periwinkle}f'}(x_{0})={\frac {1}{4}}}$

${\displaystyle {\color {Salmon}(f^{-1})'}({\color {Blue}f}(x_{0}))=4~}$