Problem

(a) Show that $ f(x) = x + r^x $ is one-to-one. …

02:38

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

in this public park every 1st 1 show that at the worst time back to be won over 100 years off. Ex found me know that off. Been worse off eggs. It's a function is one tour is equal X. He detected evidence off both sides with respect to X. You see that F prime off in search of X times. They're a joke in your function. Clams F in worse prime of X. Is it one from this three diesel issue that f commerce prime of eggs is equal to than one over f crime off a humorous off X. And that is same as what is given in the problem statement. All right, that was part of it in part B. Where has to find the worst time? Five. So we're gonna use the question that we just drive some F universe Primer five is equal to one over. Prime asked in Worse off fire. What? Has that been worse? A fire. Well, you know that effort for is it for if 405 it means that you're worse off 54 So then we have on over prior or four and primal forces, Given that history, he said. It is over to over three, so we find the answer to be then three or two.

Doruk I.

Related Courses

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Calculus: Early Transcendentals

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Implicit Differentiation

Problem

(a) Show that $ f(x) = x + r^x $ is one-to-one. …

Problem 77 Hard Difficulty

Answer

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

More Answers

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Calculus: Early Transcendentals

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

Calculus: Early Transcendentals

Catherine R.

Anna Marie V.

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Catherine R.

Anna Marie V.

Caleb E.

Michael J.

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