Derivative of inverse function/Square root/Example

The function

$f^{-1}:{ℝ}_{+}⟶{ℝ}_{+},x⟼\sqrt{x},$

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

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

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