Corollary

If $f$ is differentiable on $I$ and if $f'(x) \neq 0$ on $I$, then $g$ is differentiable on $J$ and its derivative at each point $y \in J$ is $1/f'(g(y))$.

For example, for each integer $n > 0$, the function $x^{1/n}$ is a homeomorphism of $\mathbf{R}_+$ onto itself, is the inverse of $x^n$, and has derivative $\frac{1}{n} x^{\frac{1}{n} - 1}$ at each $x > 0$.

One deduces easily, from prop. 5, that for every rational number $r = p/q > 0$ the function $x^r = (x^{1/q})^p$ has derivative $r{x'}^{-1}$ at every $x > 0$.

$^3$ For $f$ to be a homeomorphism of $I$ onto a subset of $\mathbf{R}$ we know that it is necessary and sufficient that $f$ be continuous and strictly monotone on $I$ (Gen. Top., IV, p. 338, th. 5).