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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 4.5, Problem 60E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding Second Derivatives In Exercises 57-62, find the second derivative of the function.

f ( x ) = ln x x 3 + 4 x 2

To determine

To calculate: The value of second order derivative of the function f(x)=lnxx3+4x2.

Explanation

Given information:

The provided function is f(x)=lnxx3+4x2.

Formula used:

Product rule of derivative of differentiable functions, f(x) and g(x) is:

ddx[f(x)⋅g(x)]=f′(x)⋅g(x)+f(x)⋅g′(x).

Let u be a differentiable function of x then, ddx[lnx]=1x,x>0 and ddx(lnu)=1ududx,u>0.

The derivative of function f(x)=un using the chain rule is:

f′(x)=ddx(un)=nun−1dudx

Where, u is the function of x.

Calculation:

Consider the function,

f(x)=lnxx3+4x2

Apply the property of exponential function to the function f(x)=lnxx3+4x2,

f(x)=lnxx3+4x2=x−3(lnx)+4x2

Apply the product rule of derivative to the function (x−3lnx),

The first order derivative of function f(x)=lnxx3+4x2 is,

f′(x)=ddx((x−3lnx)+4x2)=ddx(x−3lnx)+ddx(4x2)=(x−3ddx(lnx)+lnxddx(x−3))<

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