Chapter 9.7, Problem 10E

### 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 ( w ) = 1 + w 2 − w 4 1 + w 4

To determine

To calculate: The simplified form of the derivative of C(w)=1+w2w41+w4.

Explanation

Given Information:

The function is C(w)=1+w2âˆ’w41+w4.

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 quotient rule, if f(x)=u(x)v(x), then,

fâ€²(x)=uâ€²(x)v(x)âˆ’vâ€²(x)u(x)[v(x)]2

The derivative of a constant value, k, is

ddx(k)=0

Calculation:

Consider the provided function,

C(w)=1+w2âˆ’w41+w4

Differentiate both sides with respect to w,

Câ€²(w)=ddw(1+w2âˆ’w41+w4)

Apply the quotient rule to differentiate the function,

Câ€²(w)=(ddw(1+w2âˆ’w4))â‹…(1+w4)âˆ’(ddw(1+w4))â‹…(1+w2âˆ’w4)(1+w4)2=(ddw(1)+ddw(w2)âˆ’ddw(w4))â‹…(1+w4)âˆ’(ddw(1)+ddw(w4))â‹…(1+w2âˆ’w4)(1+w4)2

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

Câ€²(w)=(

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