Differentiable function/D in R/Inverse function/Fact

Derivative of inverse function

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

$f:D⟶E\subseteq ℝ$

be a bijective continuous function, with the inverse function

$f^{-1}:E⟶D.$

Suppose that $f$ is

differentiable in $a\in D$ with $f^{\prime }(a)\ne 0$.

Then also the

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

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

holds.

Proof, Write another proof