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

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

${\displaystyle f'(x)=\tan \alpha =\tan(90^{\circ }-\beta )={\frac {1}{\tan \beta }}={\frac {1}{(f^{-1})'(f(x))}}}$