Question

If $f(x)=3+x+e^{x},$ find $\left(f^{-1}\right)^{\prime}(4)$

   If $f(x)=3+x+e^{x},$ find $\left(f^{-1}\right)^{\prime}(4)$

Calculus

James Stewart 8th Edition

Chapter 6, Problem 105

Instant Answer

Solved by Expert Yiyang Wang

04/02/2021

Step 1

Step 1: We know that the derivative of the inverse function is given by the formula $\left(f^{-1}\right)^{\prime}(x) = \frac{1}{f^{\prime}(f^{-1}(x))}$. Show more…

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If $f(x)=3+x+e^{x},$ find $\left(f^{-1}\right)^{\prime}(4)$

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Key Concepts

Derivative Computation

Computing derivatives is a fundamental skill in calculus that involves applying rules like the sum rule, product rule, and chain rule. This concept is critical when working with functions composed of various terms, including elementary functions such as polynomials and exponentials, as it lays the groundwork for more advanced topics like finding the derivative of inverse functions.

Inverse Function Theorem

The inverse function theorem provides a method to determine the derivative of an inverse function without explicitly finding the inverse. It states that if a function is differentiable and its derivative is non-zero at a point, then the inverse function is locally differentiable at the corresponding point, and its derivative is simply the reciprocal of the derivative of the original function evaluated at the inverse image.

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