Derivative of inverse function/Third root/Example

The function

$f^{-1}\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{\frac {1}{3}},$

is the inverse function of the function ${}f$, given by ${}f(x)=x^{3}$. The derivative of ${}f$ in ${}a$ is ${}f'(a)=3a^{2}$, which is different from ${}0$ for ${}a\neq 0$. Due to fact, we have for ${}b\neq 0$ the relation

${}{\left(f^{-1}\right)}'(b)={\frac {1}{f'(f^{-1}(b))}}={\frac {1}{3{\left(b^{\frac {1}{3}}\right)}^{2}}}={\frac {1}{3}}b^{-{\frac {2}{3}}}\,.$

In the zero point, however, ${}f^{-1}$ is not differentiable.