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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 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 ) = 2 x 5 x 3 / 4

To determine

To find: The first and second derivative of function f(x)=2x5x34.

Explanation

Derivative Rule:

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

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

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

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

Calculation:

Obtain the first derivative of f(x).

f(x)=ddx(f(x)) =ddx(2x5x34) 

Apply the difference rule (3) and the constant multiple rule (4),

 f(x)=ddx(2x)ddx(5x34)  =2ddx(x)5ddx(x34)  

Apply the power rule (2) and simplify the terms,

 f(x)=2(1x11)5(34x341)  =2(1)5(34x3444)=25(34x14)=2154x14

Thus, the first derivative of the function f(x)=2x5x34 is f(x)=2154x14_ .

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

f(x)=ddx(f(x)) =ddx(2154x14) 

Apply the difference rule (3) and the constant multiple rule (4),

f(x)=ddx(2)ddx(154x14)=ddx(2)154ddx(x14)

Apply the derivative rule (1),

 f(x)=0ddx(15

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