   Chapter 3.2, Problem 42E ### 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

# If g(x) = x/ex. find g(n)(x).

To determine

To find: The value of gn(x).

Explanation

Given:

The function g(x)=xex.

Derivative rules:

(1) Quotient Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]f1(x)ddx[f2(x)][f2x]2

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

(3) Difference rule: ddx(fg)=ddx(f)ddx(g)

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

Calculation:

The first derivative of g(x) is g(x), which is obtained as follows.

g(x)=ddx(g(x)) =ddx(xex)

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

g(x)=(ex)ddx(x)(x)ddx(ex)(ex)2

Apply the derivative rules (2) and (4) then simplify the terms,

g(x)=(ex)(1)(x)(ex)(ex)2=exxexexex=ex(1x)exex=1xex

Thus, the first derivative of the function g(x)=xex is g(x)=1xex.

The second derivative of g(x) is g(x), which is obtained as follows,

g(x)=ddx(g(x)) =ddx(1xex)

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

g(x)=(ex)ddx(1x)(1x)ddx(ex)(ex)2

Apply the derivative rule (3),(2) and (4),

g(x)=(ex)[ddx(1)ddx(x)](1x)ddx(ex)(ex)2=((ex)(01))((1x)(ex))(ex)2=ex(exxex)(ex)2=2ex+xex(ex)2

Divide the numerator and the denominator by ex

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