   Chapter 4.5, Problem 59E ### 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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# Finding Second Derivatives In Exercises 57-62, find the second derivative of the function. f ( x ) = 5 + ln x 3 x 2

To determine

To calculate: The value of second order derivative of the function f(x)=5+lnx3x2.

Explanation

Given information:

The provided function is f(x)=5+lnx3x2.

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)=nun1dudx

Where, u is the function of x.

Calculation:

Consider the function,

f(x)=5+lnx3x2

Apply the property of exponential function to the provided function f(x)=5+lnx3x2,

f(x)=5+lnx3x2=5+13x2(lnx)

Apply the product rule of derivative to (x2lnx),

The first order derivative of function f(x)=5+lnx3x2 is,

f(x)=ddx(5+13(x2lnx))=ddx(5)+ddx13(x2lnx)=0+13(x2ddx(lnx)+lnxddx(x2))=13(x2(1x)+lnx(2x21))</

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