Inverse function theorem

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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'(a) is nonzero, i.e., f'(a) \ne 0. Then:

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

(f^{-1})'(b) = \frac{1}{f'(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})'(x) = \frac{1}{f'(f^{-1}(x))}

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

One-sided versions

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Infinity-sensitive versions

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