Second derivative rule for inverse function

This article is about a differentiation rule, i.e., a rule for differentiating a function expressed in terms of other functions whose derivatives are known.
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Statement

Simple version at a specific point

Suppose $f$ is a one-one function and $a$ is a point in the domain of $f$ such that $f$ is twice differentiable at $a$ and $f^{\prime }(a)\ne 0$ where $f^{\prime }$ denotes the derivative of $f$. Suppose $b=f(a)$.

Then, we have the following formula for the second derivative of the inverse function $f^{-1}$:

$(f^{-1}{)}^{″}(b)=\frac{-f^{″}(a)}{(f^{\prime }(a){)}^3}$

Simple version at a generic point

Suppose $f$ is a one-one function. Then, we have the following formula:

$(f^{-1}{)}^{″}(x)=\frac{-f^{″}(f^{-1}(x))}{(f^{\prime }(f^{-1}(x)){)}^3}$

where the formula is applicable for all $x$ in the range of $f$ for which $f$ is twice differentiable at $f^{-1}(x)$ and the first derivative of $f$ at $f^{-1}(x)$ is nonzero.