# The derivative of the function F ( x ) = x 4 − 5 x 3 + x x 2 . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.2, Problem 2E
To determine

## To find: The derivative of the function F(x)=x4−5x3+xx2.

Expert Solution

The derivative of the function F(x)=x45x3+xx2 is. 2x532x52.

### Explanation of Solution

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)](x45x3+x12)2xx4=x2[4x315x2+12x12](x45x3+x12)2xx4=4x515x4+12x2122x5+10x42x1+12x4=4x515x4+12x322x5+10x42x32x4

Simplify the numerator and obtain the differentiation of F(x).

F(x)=2x55x432x32x4=2x532x324=2x532x52

Therefore, the differentiation of the function F(x)=x45x3+xx2 is 2x532x52.

Method 2:

Obtain the derivative of F(x) by simplifying first.

Simplify the function F(x),

F(x)=x45x3+x12x2=x4x25x3x2+x12x2=x25x+x122=x25x+x32

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

F(x)=ddx(F(x))=ddx(x25x+x32)

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

F(x)=ddx(x2)ddx(5x)+ddx(x32)=ddx(x2)5ddx(x)+ddx(x32)

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

F(x)=2x5(1)32x321=2x532x52

Thus, the differentiation of the function F(x)=x45x3+xx2 is 2x532x52.

Therefore, it can be concluded that the derivative of F(x) by both the methods 1 and 2 are same and method 2 seems to be the better one as it has lesser calculations.

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