Proposition: Derivative of an Invertible Function on Real Invervals

Let \([a,b]\) be a closed real interval and let \(f:[a,b]\to\mathbb R\) be a continuous, strictly monotonically increasing (respectively decreasing) real function, and let $\phi:=f^*:D\to\mathbb R$ be its inverse function, where $D=f^*([a,b]).$ If $f$ is differentiable at a point $x\in[a,b]$ and if $f(x)\neq 0$, then the function $\phi$ is differentiable at a point $y=f(x).$ In particular, $$\phi'(x)=\frac1{f'(x)}=\frac1{f'(\phi(y))}.$$

Proofs: 1

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983