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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable.

f(x) = x5 – 2x3 + x – 1

To determine

To find: The derivative of f(x). That is f(x).

Explanation

Given:

The function is f(x)=x52x3+x1

Derivative rules:

(1) Derivative of constant function: ddx(c)=0

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

(3) Sum Rule: ddx(f+g)=ddx(f)+ddx(g)

(4) Difference Rule: ddx(fg)=ddx(f)ddx(g)

(5) Constant Multiple Rule: ddx(cf)=cddx(f)

Calculation:

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

f(x)=ddx(f(x)) =ddx(x52x3+x1) 

Apply the sum rule (3) and the difference rule (4),

 f(x)=ddx(x52x3)+ddx(x1) =ddx(x5)ddx(2x3)+ddx(x)ddx(1)

Apply the derivative rule (5) and derivative rule (1),

 f(x)=ddx(x5)2ddx(x3)+ddx(x)ddx(1)

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