   Chapter 9.8, Problem 24E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In Problems 13-24, find the indicated derivative.Find f ( 3 ) ( x ) if f ' ( x ) = x 2 x 2 + 1

To determine

To calculate: The value of f(3)(x) if the derivative for the function is f(x)=x2x2+1.

Explanation

Given Information:

The provided derivative for the function is f(x)=x2x2+1.

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)

The derivative of a constant value, k, is

ddx(k)=0

According to the quotient rule of derivative, if there is a function of form y=fg, then its derivative is

dydx=gffgg2

According to the chain rule of derivatives, if there is a function of form y=f(u(x)), then its derivative is

dydx=f(u(x))u(x)

Calculation:

Consider the provided derivative for the function,

f(x)=x2x2+1

To get the value of f(2)(x), differentiate both sides of f(x) with respect to x,

f(2)(x)=ddx(x2x2+1)

Apply the quotient and power rule of derivatives,

f(2)(x)=(x2+1)(ddx(x2))(x2)(ddx(x2+1))(x2+1)2=(x2+1)(2x21)(x2)(ddx(x2)+ddx(1))(x2+1)2=(x2+1)(2x)(x2)(2x21+0)(x2+1)2=2x(x2+1)2x3(x2+1)2

Simplify it further,

f(2)(x)=2<

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