   Chapter 3.2, Problem 28E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find f'(x) and f"(x). f ( x ) = x e x

To determine

To find: The derivatives f(x) and f(x).

Explanation

Given:

The function f(x)=xex.

Derivative rule:

(1) Product Rule: ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)]

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

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

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

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

Calculation:

The first derivative of the function f(x)=xex is f(x), which is obtained as follows,

f(x)=ddx(f(x))=ddx(xex)=ddx(x12ex)

Apply the product rule (1),

f(x)=(x12)ddx(ex)+exddx(x12)

Apply the derivative rules (2),(5) and simplify the terms.

f(x)=(x12)(ex)+ex(12x121)=(x12)(ex)+ex(12x1222)=exx12+12exx12=ex(x12+12x12)

Therefore, the first derivative of the function f(x)=xex is f(x)=ex(x12+12x12)

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