   Chapter 3.2, Problem 27E ### 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) = (x3 + 1)ex

To determine

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

Explanation

Given:

The function f(x)=(x3+1)ex.

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)=(x3+1)ex is f(x), which is obtained as follows,

f(x)=ddx(f(x))=ddx((x3+1)ex)

Apply the product rule (1) and simplify the terms,

f(x)=(x3+1)ddx(ex)+exddx(x3+1)

Apply the derivative rules (3),(2) and (5),

f(x)=[(x3+1)ddx(ex)]+ex[ddx(x3)+ddx(1)]=(x3+1)ex+ex(3x31+0)=(x3+1)ex+ex(3x2)=ex(x3+3x2+1)

Therefore, the first derivative of the function f(x)=(x3+1)ex is f(x)=ex(x3+3x2+1)

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