   Chapter 9, Problem 74RE 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 S' if S = ( 3 x + 1 ) 2 x 2 − 4

To determine

To calculate: The simplified form of the derivative of S=(3x+1)2x24.

Explanation

Given Information:

The function is S=(3x+1)2x24.

Formula used:

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

f(x)=nxn1

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

g(x)=cf(x)

According to the sum rule 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 chain rule differentiation, if y=f(u), where u=g(x), then y is a differentiable function of x.

dydx=dydududx

Calculation:

Consider the provided function,

S=(3x+1)2x24

Consider (3x+1) to be u,

S=u2x24

Differentiate both sides with respect to x,

S=ddx(u2x24)

Simplify by the use of the quotient rule,

dCdx=(ddx(u2))(x24)(ddx(x24))(u2)(x24)2=(ddx(u2))(x24)(ddx(x2)ddx(4))(u2)(x24)2

Simplify by the use of the power rule,

y=(2u21dudx)(x24)((2x21)0)(u2)(x24)2=(2ududx)(x24

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