BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.4, Problem 60E

a.

To determine

Find theexpression for F'(x) using given data.

Expert Solution

Answer to Problem 60E

The expression is F'(x)=f'(xα)αxα1 .

Explanation of Solution

Given:

The given datais F(x)=f(xα) , f is differentiable on R .

Calculation:

  F(x)=f(xα)

Apply chain rule.

  df(a)dx=dfdadadx

  F'(x)=dd(xα)[f(xα)]ddx(xα)

Use derivative rule.

  ddx(xn)=nxn1

  F'(x)=f'(xα)αxα1

Hence theexpression is F'(x)=f'(xα)αxα1 .

b.

To determine

Find the expression for G'(x) using given data.

Expert Solution

Answer to Problem 60E

The expression is G'(x)=α[f(x)]α1f'(x) .

Explanation of Solution

Given:

The given data is G(x)=[f(x)]α , f is differentiable on R .

Calculation:

  G(x)=[f(x)]α

Apply chain rule.

  df(a)dx=dfdadadx

Let f=aαanda=f(x)

  G'(x)=dda[aα]ddx[f(x)]

Use derivative rule.

  ddx(xn)=nxn1

  G'(x)=αaα1f'(x)

Substitute the value of a=f(x) .

  G'(x)=α[f(x)]α1f'(x)

Hence the expression is G'(x)=α[f(x)]α1f'(x) .

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