   Chapter 3.2, Problem 64E

Chapter
Section
Textbook Problem

(a) If g is differentiable, the Reciprocal Rule says that d d x [ 1 g ( x ) ] = − g ' ( x ) [ g ( x ) ] 2 Use the Quotient Rule to prove the Reciprocal Rule.(b) Use the Reciprocal Rule to differentiate the function in Exercise 16.(c) Use the Reciprocal Rule to verify that the Power Rule is valid for negative integers, that is, d d x ( x − n ) = − n x − n − 1 for all positive integers n.

(a)

To determine

To show: The reciprocal rule ddx[1g(x)]=g(x)[g(x)]2.

Explanation

Given:

The function g(x) is differentiable.

Derivative rules:

(1) Quotient Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]f1(x)ddx[f2(x)][f2x]2

(2) Power rule: ddx(xn)=nxn1

(3) Sum rule: ddx(f+g)=ddx(f)+ddx(g)

(4) Difference rule: ddx(fg)=ddx(f)</

(b)

To determine

To find: Differentiation of the function by using the reciprocal rule.

(c)

To determine

To verify: The power rule is valid for negative integer. That is, ddx(xn)=nxn1

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

Solve the equations in Exercises 112 for x (mentally, if possible). x3=1

Finite Mathematics and Applied Calculus (MindTap Course List)

Find the indicated power using De Moivres Theorem. 35. (23+2i)5

Single Variable Calculus: Early Transcendentals, Volume I

Fill in the blanks. 8. a. The general form of an equation of a line is_______.

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Factor: 3x26x189

Elementary Technical Mathematics

0 2 −2 does not exist

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

True or False: Du f(a, b) = ∇(a, b) · u.

Study Guide for Stewart's Multivariable Calculus, 8th 