   Chapter 3.1, Problem 48E

Chapter
Section
Textbook Problem

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of f, f', and f". f ( x ) = e x − x 3

To determine

To find: The first and second derivative of function.

Explanation

Given:

The function f(x)=exx3.

Derivative Rule:

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

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

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

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

Calculation:

Obtain the first derivative of f(x).

f(x)=ddx(f(x))=ddx(exx3)

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

f(x)=ddx(ex)ddx(x3) =exddx(x3)

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

f(x)=ex(3x31)=ex3x2

Thus, the first derivative of the function f(x)=exx3 is f(x)=ex3x2_.

Obtain the second derivative of the function f(x).

f(x)=ddx(f(x)) =ddx(ex3x2)

Apply the difference rule (2) and the constant multiple rule (3),

f(x)=ddx(ex)ddx(3x2)=ddx(

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

Evaluate the integrals in Problems 1-32. 26

Mathematical Applications for the Management, Life, and Social Sciences

For

Study Guide for Stewart's Multivariable Calculus, 8th

The slope of the tangent line at the point where to the curve y = sin 2t, x = cos 3t is: 3 −3

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

In Exercise 118, construct a truth table for each compound proposition. (pq)(pr)

Finite Mathematics for the Managerial, Life, and Social Sciences 