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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 derivative o f the function

F ( x ) = x 4 5 x 3 + x x 2

in two ways: by using the Quotient Rule and by simplifying first. Show that your answers are equivalent. Which method do you prefer?

To determine

To find: The derivative of the function F(x)=x45x3+xx2.

Explanation

Given:

The function F(x)=x45x3+xx2.

Derivative rule:

(1) Quotient Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]f1(x)ddx[f2(x)][f2(x)]2

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

(3). Sum rule: ddx(f+g)=ddx(f)+ddx(g)

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

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

Calculation:

Method 1:

Obtain the derivative of F(x) by using Quotient Rule.

The derivative of the function F(x) is F(x), which is obtained as follows.

F(x)=ddx(F(x))=ddx(x45x3+xx2)

Substitute x45x3+x for f1(x) and x2 for f2(x) in the quotient rule (1),

F(x)=(x2)ddx(x45x3+x)(x45x3+x)ddx(x2)(x2)2

Apply the derivative rules (3), (4), and (5),

F(x)=(x2)[ddx(x4)ddx(5x3)+ddx(x12)][(x45x3+x)ddx(x2)](x2)2=x2[ddx(x4)5ddx(x3)+ddx(x12)][(x45x3+x12)ddx(x2)]x4

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

F(x)=x2[4x35(3x31)+(12x121)](x4−</

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