Derivative of inverse function/Square root/Example

The function

$f^{-1}\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} _{+},x\longmapsto {\sqrt {x}},$

is the inverse function of the function ${}f$, given by ${}f(x)=x^{2}$ (restricted to ${}\mathbb {R} _{+}$). The derivative of ${}f$ in a point ${}a$ is ${}f'(a)=2a$. Due to fact, for ${}b\in \mathbb {R} _{+}$, the relation

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

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