Question

Find a formula for $\left(f^{-1}\right)^{\prime}(x)$ given that $f$ is one-to-one and its derivative satisfies the equation given. $$f^{\prime}(x)=\sqrt{1-[f(x)]^{2}}$$

Name: Problem 47, Chapter 7: Calculus: One Variable
Uploaded: 2021-11-04T15:33:39-08:00
Duration: 1 min 46 s
Channel: Nick Johnson
Description: Find a formula for (f^-1)^'(x) given that f is one-to-one and its derivative satisfies the equation given. f^'(x)=√(1-[f(x)]^2)

   Find a formula for $\left(f^{-1}\right)^{\prime}(x)$ given that $f$ is one-to-one and its derivative satisfies the equation given.
$$f^{\prime}(x)=\sqrt{1-[f(x)]^{2}}$$

Calculus: One Variable

Garret J. Etgen,… 10th Edition

Chapter 7, Problem 47

Instant Answer

Solved by Expert Nick Johnson

11/04/2021

Step 1

If $ y = f^{-1}(x) $, then \[ (f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}. \] We need to find $ f'(f^{-1}(x)) $ to use this formula. Show more…

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Find a formula for $\left(f^{-1}\right)^{\prime}(x)$ given that $f$ is one-to-one and its derivative satisfies the equation given. $$f^{\prime}(x)=\sqrt{1-[f(x)]^{2}}$$

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Key Concepts

Inverse Function Derivative

This concept states that if a function f is one-to-one and differentiable, then its inverse function is differentiable wherever f' is non-zero. The derivative of the inverse function is given by (f?¹)'(x) = 1 / f'(f?¹(x)), which provides a method to compute the derivative of the inverse directly in terms of the derivative of the original function.

Chain Rule

The chain rule is fundamental in differentiating composite functions. In the context of inverse functions, it is applied to differentiate the composition f(f?¹(x)) and leads to the relation f'(f?¹(x)) * (f?¹)'(x) = 1. This step is essential in deriving the formula for the derivative of the inverse function.

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