   Chapter 3.5, Problem 77E

Chapter
Section
Textbook Problem

(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 ) = 1 f ' ( f − 1 ( x ) ) provided that the denominator is not 0.(b) If f(4) = 5 and f ' ( 4 ) = 2 3 , find (f–1)′(5).

(a)

To determine

To show: The derivative of the inverse function (f1)(x)=1f(f1(x)).

Explanation

Given:

The function f(x) is one to one and differentiable and its inverse f1 also differentiable.

Derivative rules:

Chain rule: dydx=dydududx

Proof:

Obtain the derivative of the function f1(x).

Since the function f(x) is one to one function, f(f1(x))=x.

Let u=f1(x).

f(u)=x

Differentiate implicitly with respect to x.

ddx(f(u))=ddx(x)ddx(f(u))=1

Apply the chain rule and simplify the terms,

ddu(f(u))

(b)

To determine

To find: The value f1(5).

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts 