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Numerade Educator

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Problem 77 Hard Difficulty

(a) Suppose $ f $ is a one-to-one differentiable function and its inverse function $ f^{-1} $ is also differentiable. Use implicit differentiation to show that
$ (f^{-1})'(x) = \frac {1}{f'(f^{-1}(x))} $
provided that the denominator is not 0.
(b) If $ f(4) = 5 $ and $ f'(4) = \frac {2}{3}, $ find $ (f^{-1})'(5). $

Answer

(a) $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$
(b) $$ \frac{3}{2} $$

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Video Transcript

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