Chapter 9.7, Problem 24E

### 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 = ( 5 − x 2 x 4 ) 3

To determine

To calculate: The simplified form of the derivative of y=(5x2x4)3.

Explanation

Given Information:

The provided function is y=(5âˆ’x2x4)3.

Formula used:

According to the powers of x rule, if f(x)=xn, then,

fâ€²(x)=nxnâˆ’1

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

gâ€²(x)=cfâ€²(x)

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

fâ€²(x)=uâ€²(x)âˆ’vâ€²(x)

According to the difference rule, if a function is of the form f(x)=u(x)âˆ’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

According to the chain rule, if a function is of the form y=un, where u=g(x),

dydx=nunâˆ’1dudx

Calculation:

Consider the provided function,

y=(5âˆ’x2x4)3

Let, (5âˆ’x2x4) to be u,

y=u3

Differentiate both sides with respect to x,

yâ€²=ddx(u3)

Simplify by the use of power of x rule,

yâ€²=3â‹…u3âˆ’1â‹…dudx=3u2dudx

Substitute (5âˆ’x2x4) for u,

yâ€²=3(5âˆ’x2x4)2ddx(5âˆ’x2x4)

Simplify the internal derivative ddx(5âˆ’x2x4) by the use of quotient rule and the difference rule,

yâ€²=3(5âˆ’x2x4)2((ddx(5âˆ’x2))â‹…(x4)âˆ’(ddx(x4))â‹…(5âˆ’x2)(x

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