   Chapter 3.1, Problem 29E ### 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

# Differentiate the function. f ( v ) = v 3 − 2 v e v v

To determine

To find: The derivative of the function f(v)=v32vevv.

Explanation

Given:

The function, f(v)=v32vevv.

Formula used:

The Constant Multiple Rule:

If c is a constant and f(v) is a differentiable function, then the constant multiple rule is,

ddv[cf(v)]=cddvf(v) (1)

The Power Rule:

If n is any real number, then the power rule is,

ddv(vn)=nvn1 (2)

Derivative of the Natural Exponential Function:

ddu(eu)=eu (3)

The Difference Rule:

If f(v) and g(v) are both differentiable functions, then the difference rule is,

ddv[f(v)g(v)]=ddv(f(v))ddv(g(v)) (4)

Calculation:

The derivative of f(v)=v32vevv is f(v), which is obtained below:

f(v)=ddv(f(v)) =ddv(v32vevv)=ddv(v132vevv)=ddv(v13v2vevv)

f(v)=ddv(v13v2ev)=ddv(v1312ev)

Apply the difference rule as shown in equation (4).

f(v)=ddv(v131)ddv(2ev)

Apply the constant multiple rule as shown in equation (1)

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