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

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
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,

f(x)=nxn1

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

g(x)=cf(x)

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

f(x)=u(x)+v(x)

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

f(x)=u(x)v(x)+v(x)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=nun1dudx

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,

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

Simplify by the use of the product rule,

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

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

f(x)=((2u21dudx)(5x3+1)+(5(3x31)+0)u2)=((2ududx)(5x3+1)+(15x2)u2)

Take u common,

f(x)=u((2d

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