   Chapter 4.5, Problem 4CP ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Find the derivative of f ( x ) = ln [ x 2 x 2 + 1 ] .

To determine

To calculate: The derivative of the function f(x)=ln(x2x2+1).

Explanation

Given information:

The provided function is f(x)=ln(x2x2+1).

Formula used:

The chain rule of derivative for natural logarithm

ddxlnu=1ududx.

Where, u is the function of x.

The property of logarithms,

lnxa=alnx

The product law of logarithms,

ln(xy)=lnx+lny

Where x and y are real numbers greater than 0.

Calculation:

Consider the function,

f(x)=ln(x2x2+1)

Apply the product law of logarithms,

f(x)=ln(x2x2+1)=lnx2+ln(x2+1)12=2lnx+12ln(x2+1)

Let; u=x2+1,

Now differentiate the function with respect to x,

dudx=ddx(x2+1)=2x0=2x

Substitute, u for x2+1 in function,

f(x)=2lnx+12lnu

Now differentiate the function with respect to x

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