   Chapter 11.6, Problem 66E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

Use logarithmic differentiation to give a proof of the quotient rule.

To determine

To prove: The quotient rule ddx[f(x)g(x)]=f(x)g(x)f(x)g(x)[g(x)]2 by using logarithmic differentiation.

Explanation

Given Information:

The quotient rule is ddx[f(x)g(x)]=f(x)g(x)f(x)g(x)[g(x)]2.

Formula used:

The derivative of natural logarithm of a function is ddxlnu=1ududx.

Proof:

Consider the function, y=f(x)g(x)

Now, take natural logarithm on both sides of the function as,

lny=ln[f(x)g(x)]

Simplify the above expression as,

lny=ln[f(x)]ln[g(x)]

Then, take ddx of both sides on the above expression,

ddx(lny)=ddx(ln[f(x)]ln[g(x)])=ddx(ln[f(x)])ddx(ln[g(x)])

The derivative of natural logarithm of a function is ddxlnu=1ududx.

Apply the above formula for the derivative,

1ydydx=1f(x)ddx[f(x)]1g(x)ddx[g(x)]=f(x)f(x)

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

7. List four perils covered by property insurance. (19-4)

Contemporary Mathematics for Business & Consumers

Given:12. Prove: ABAC=BECD

Elementary Geometry For College Students, 7e

Simplify each power of i. i28

Trigonometry (MindTap Course List)

True or False: converges absolutely.

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

An iterated integral for the volume of the solid shown is:

Study Guide for Stewart's Multivariable Calculus, 8th 