Inverse function theorem

Statement

Simple version at a specific point

Suppose $f$ is a function of one variable that is a one-one function and $a$ is in the domain of $f$. Suppose $f$ is [differentiable function|differentiable]] at $a$ and $b=f(a)$. Suppose further that the derivative $f^{\prime }(a)$ is nonzero, i.e., $f^{\prime }(a)\ne 0$. Then:

The inverse function $f^{-1}$ is differentiable at $b$, and further:

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

Simple version at a generic point

Suppose $f$ is a function of one variable that is a one-one function. Then, the formula for the derivative of the inverse function is as follows:

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

with the formula applicable at all points in the range of $f$ for which $f^{\prime }(f^{-1}(x))$ exists and is nonzero.

One-sided versions

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Infinity-sensitive versions

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