Chapter 9.7, Problem 3CP

### 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 derivative of each of the following and simplify.(a) f ( x ) = 3 x 4 ( 2 x 4 + 7 ) 5 (b) g ( x ) = ( 4 x + 3 ) 7 2 x − 9

(a)

To determine

To calculate: The derivative of y=3x4(2x4+7)5.

Explanation

Given Information:

The function is y=3x4(2x4+7)5.

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,

y=3x4(2x4+7)5

Consider (2x4+7) to be u,

y=3x4u5

Differentiate both sides with respect to x,

yâ€²=ddx(3x4u5)=3ddx(x4u5)

Simplify using the product rule,

yâ€²=3((ddx(x4))â‹…u5+(ddx(u5))â‹…x4)

Simplify using the power rule,

yâ€²=3((4x4âˆ’1)â‹…u5+(5u5âˆ’1â‹…dudx)â‹…x4)=3((4x3)â‹…u5+(5u4â‹…dud

(b)

To determine

To calculate: The derivative of g(x)=(4x+3)72x9.

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