   Chapter 2.6, Problem 48E

Chapter
Section
Textbook Problem

# The Power Rule can be proved using implicit differentiation for the case where n is a rational number, n = p / q , and y = f ( x ) = x n is assumed beforehand to be a differentiable function. If y = x p / q , then y q = x p . Use implicit differentiation to show that y ′ = p q x ( p / q ) − 1

To determine

To show:

y'=pqxpq-1

Solution:In the following we prove that when y=xp/q for some integers p and q, q not equal to zero

y'=pqxpq-1

Explanation

1) Formula:

Power rule of differentiation:

ddxxn=nxn-1

2) Given:

y=xn here n=pq

If  y=xpq then yq=xp

Calculation :

Consider

y=xpq where p and q are integers with q not equal to zero.

Multiply the equation of power by q

yq=xpq*q

yq=xp

Differentiate with respect to x

ddxyq=xp

ddxyq=ddxxp

Use power rule of differnetiation combined with chain rule

qyq-1dydx=pxp-1

dydx=pxp-1qyq-1

Now as per given condition,

### 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 126. 2(x21)x2+1x4x2+1x2+1=0

Finite Mathematics and Applied Calculus (MindTap Course List)

#### In Exercises 107-120, factor each expression completely. 114. 3x2 4x 4

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

#### Is 460 divisible by3?

Elementary Technical Mathematics

#### In problems 33-36, write an inequality that discribes each interval or graph. 35.

Mathematical Applications for the Management, Life, and Social Sciences

#### If , then h′(x) = 8t3 + 2t 24x2 + 2 8x3 + 2x – 3 8x3 + 2x

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