Derivative of the inverse function

If a function is invertible, its inverse can also be differentiated: the derivative of the inverse is the reciprocal of the derivative, evaluated at the corresponding point.

Property — Derivative of the inverse function

If $f$ is a bijection and differentiable with $f'(x)\ne 0$, the derivative of the inverse is: $(f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))}.$ In Leibniz notation: $\dfrac{dx}{dy} = \dfrac{1}{dy/dx}$ — again, “like a fraction”.

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Topics: Derivatives
Concepts: Derivative · Inverse function · Leibniz notation
Skills: Differentiating