Chapter 3.2, Problem 4E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Differentiate. g ( x ) = ( x + 2 2 ) e x

To determine

To find: The differentiation of the function g(x)=(x+2x)ex.

Explanation

Given:

The function g(x)=(x+2x)ex.

Derivative rule:

(1) Product Rule:

If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)]

(2) Constant multiple rule: ddx(cf)=cddx(f)

(3) Power rule: ddx(xn)=nxnâˆ’1

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

(5) Natural exponential function: ddx(ex)=ex

Calculation:

The derivative of the function g(x) is gâ€²(x), which is obtained as follows,

gâ€²(x)=ddx(g(x))=ddx[(x+2x)ex]

Substitute x+2x for f1(x) and ex for f2(x) in the product rule (1),

gâ€²(x)=(x+2x)ddx(ex)+exddx(x+2x)=(x+2x12)ddx(ex)+exddx(x+2x12)

Apply the derivative

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