Chapter 9.7, Problem 8E

Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Chapter
Section

Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only. y = ( x 3 − 5 x 2 + 1 ) ( x 3 − 3 )

To determine

To calculate: The simplified form of the derivative of y=(x35x2+1)(x33).

Explanation

Given Information:

The function is y=(x3âˆ’5x2+1)(x3âˆ’3).

Formula used:

According to the power rule, if f(x)=xn, then,

fâ€²(x)=nxnâˆ’1

According to the property of differentiation, if a function is of the form, g(x)=cf(x), then,

gâ€²(x)=cfâ€²(x)

According to the product rule, if f(x)=u(x)â‹…v(x), then

fâ€²(x)=uâ€²(x)â‹…v(x)+vâ€²(x)â‹…u(x)

According to the property of differentiation, if a function is of the form f(x)=u(x)+v(x), then,

fâ€²(x)=uâ€²(x)+vâ€²(x)

The derivative of a constant value, k, is

ddx(k)=0

Calculation:

Consider the provided function,

y=(x3âˆ’5x2+1)(x3âˆ’3)

Differentiate both sides with respect to x,

yâ€²=ddx((x3âˆ’5x2+1)(x3âˆ’3))

Simplify using the product rule,

yâ€²=(ddx(x3âˆ’5x2+1))â‹…(x3âˆ’3)+(ddx(x3âˆ’3))â‹…(x3âˆ’5x2+1)=(ddx(x3)âˆ’ddx(5x2)+ddx(1))â‹…(x3âˆ’3)+(ddx(x3)âˆ’ddx(

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