Chapter 9.7, Problem 31E

### 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. c ( x ) = 2 x x 3 + 1

To determine

To calculate: The simplified form of the derivative of c(x)=2xx3+1.

Explanation

Given Information:

The function is c(x)=2xx3+1.

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,

c(x)=2xx3+1

Rewrite the function,

c(x)=2x(x3+1)12

Consider (x3+1) to be u,

c(x)=2xu12

Differentiate both sides with respect to x,

câ€²(x)=ddx(2xu12)=2ddx(xu12)

Simplify using the product rule,

câ€²(x)=2((ddx(x))â‹…u12+(ddx(u12))â‹…x)

Simplify using the power rule,

câ€²(x)=2((x1âˆ’1)â‹…u12+(12u12âˆ’1â‹…dudx)â‹…x)=2(u12+(12uâˆ’12â‹…ddx(u))â‹…x)

Take u12 common,

câ€²(x)=2u12(1</

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