   Chapter 3.2, Problem 42E

Chapter
Section
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

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Solve the equations in Exercises 126. x43x3=0

Finite Mathematics and Applied Calculus (MindTap Course List)

[(13)2]3

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

In problems 23-58, perform the indicated operations and simplify. (x3+5x26)(x22)

Mathematical Applications for the Management, Life, and Social Sciences

True or False: , for –π ≤ t ≤ π, is smooth.

Study Guide for Stewart's Multivariable Calculus, 8th 