   Chapter 3.2, Problem 62E ### 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

# (a) If F(x) = f(x) g(x), where f and g have derivatives of all orders, show that F" = f"g + 2f'g' + fg".(b) Find similar formulas for F"' and F(4).(c) Guess a formula for F(n).

(a)

To determine

To show: If F(x)=f(x)g(x), then F=fg+2fg+fg.

Explanation

Derivative rule:

(1) Product Rule: ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)]

(2) Sum Rule: ddx[f1(x)+f2(x)]=ddx[f1(x)]+ddx[f2(x)]

Proof:

Obtain the first derivative of the function F(x).

F(x)=ddx(f(x)g(x))

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

F(x)=f(x)ddx(g(x))+g(x)ddx(f(x))=f(x)g(x)+g(x)f(x)

Thus, the first derivative of the function F(x) is F(x)=f(x)g(x)+g(x)f(x).

Obtain the second derivative of the function F(x)=f(x)g(x)

(b)

To determine

To find: The formulas for f(x) and f4(x).

(c)

To determine

To guess: The formula for F(n).

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