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Chapter 9.7, Problem 16E
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Mathematical Applications for the ...

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

Solutions

Chapter
Section
BuyFindarrow_forward

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. f ( x ) = ( 5 x 3 + 1 ) ( x 4 + 5 x ) 2

To determine

To calculate: The simplified form of the derivative of f(x)=(5x3+1)(x4+5x)2.

Explanation

Given Information:

The function is f(x)=(5x3+1)(x4+5x)2.

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 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)

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)

The derivative of a constant value, k, is

ddx(k)=0

According to the property of differentiation, if a function is of the form y=un, where u=g(x),

dydx=nunâˆ’1dudx

Calculation:

Consider the provided function,

f(x)=(5x3+1)(x4+5x)2

Consider (x4+5x) to be u,

f(x)=(5x3+1)u2

Differentiate both sides with respect to x,

fâ€²(x)=ddx((5x3+1)u2)

Simplify by the use of the product rule,

fâ€²(x)=((ddx(u2))â‹…(5x3+1)+(ddx(5x3+1))â‹…u2)=((ddx(u2))â‹…(5x3+1)+(5ddx(x3)+ddx(1))â‹…u2)

Simplify by the use of the power rule and the rule for constants,

fâ€²(x)=((2u2âˆ’1â‹…dudx)â‹…(5x3+1)+(5(3â‹…x3âˆ’1)+0)â‹…u2)=((2uâ‹…dudx)â‹…(5x3+1)+(15x2)â‹…u2)

Take u common,

fâ€²(x)=u((2d

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