Chapter 9.7, Problem 15E

### 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 − 1 ) 2 ( x 2 + 1 )

To determine

To calculate: The simplified form of the derivative of y=(x1)2(x2+1).

Explanation

Given Information:

The function is y=(xâˆ’1)2(x2+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,

y=(xâˆ’1)2(x2+1)

Consider (xâˆ’1) to be u,

y=u2(x2+1)

Differentiate both sides with respect to x,

yâ€²=ddx(u2(x2+1))

Simplify using the product rule,

yâ€²=((ddx(u2))â‹…(x2+1)+(ddx(x2+1))â‹…u2)=((ddx(u2))â‹…(x2+1)+(ddx(x2)+ddx(1))â‹…u2

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