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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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(

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