   Chapter 11, Problem 3T ### 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 1-8, find the derivative of each function. 3.   y = ln   ( x 4 + 1 ) 3

To determine

To calculate: The derivative of the function y=ln(x4+1)3.

Explanation

Given Information:

The provided function is,

y=ln(x4+1)3

Formula used:

Property of logarithms:

ln(Mp)=plnM

The differential ddx(cf(x)) can be written as cddxf(x) where c is any non-zero real number.

If y=lnu, where u is a differentiable function of x, then

dydx=1ududx

For functions f(x) and g(x),

ddx[f(x)±g(x)]=ddxf(x)±ddxg(x)

According to the power rule of derivatives:

ddx(xn)=nxn1

Calculation:

Consider the provided function:

y=ln(x4+1)3

First, take the derivative of both sides of the equation with respect to x as:

dydx=ddx[ln(x4+1)3]

Use the property of logarithms ln(Mp)=plnM. Then,

dydx=ddx[ln(x4+1)3]=ddx[3ln(x4+1)]

Now, for any non-zero real number c, ddx(cf(x))=cddxf(x).

dydx=ddx[3ln(x4+1)]=3ddx[ln(x4+1)]

Now, if y=lnu, where u is a differentiable function of x, then dydx=1ududx

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