   Chapter 9.7, Problem 18E Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Solutions

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 2 − 3 ) 4 x

To determine

To calculate: The simplified form of the derivative of y=(x23)4x.

Explanation

Given Information:

The function is y=(x23)4x.

Formula used:

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

f(x)=nxn1

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

g(x)=cf(x)

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

f(x)=u(x)+v(x)

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

f(x)=u(x)v(x)v(x)u(x)[v(x)]2

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=nun1dudx

Calculation:

Consider the provided function,

y=(x23)4x

Consider (x23) to be u,

y=u4x

Differentiate both sides with respect to x,

y=ddx(u4x)

Simplify by the use of the quotient rule,

y=(ddx(u4))(x)(ddx(x))(u4)x2

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

y=(4u41dudx)(x)(x11)(u4)x2=(4u3

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