Differentiable function/D in R/Inverse function/Fact

Derivative of inverse function

Let ${}D,E\subseteq \mathbb {R}$ denote intervals, and let

$f\colon D\longrightarrow E\subseteq \mathbb {R}$

be a bijective continuous function, with the inverse function

$f^{-1}\colon E\longrightarrow D.$

Suppose that ${}f$ is

differentiable in ${}a\in D$ with ${}f'(a)\neq 0$.

Then also the

inverse function ${}f^{-1}$ is differentiable in ${}b:=f(a)$, and

${}(f^{-1})'(b)={\frac {1}{f'(f^{-1}(b))}}={\frac {1}{f'(a)}}\,$

holds.

Proof, Write another proof