Expert: Paul Klarreich - 3/16/2008

Question
QUESTION: I am confused when it comes to finding the second derivative of a function.  An example of this is with the function:

f(x) (x^2+2)^5

I find the first derivative as
10x(x^2+2)^4
Then, in looking at the equation I figure that I need to use the product rule next.

I take the first:
(10x)
times the derivative of the second
(4)(x^2)^3 (2x)
and add that to the second:
(x^2+2)^4
times the derivative of the first:
(10)

This gives me:
(10x)(4)(x^2+2)^3(2x)+(x^2+2)^4(10)

I am unsure of how to proceed next.  The solutions manual says that the second derivative answer is:

10(9x^2+2)(x^2+2)^3??

The other problem I am working on is very similar and I felt confident until I saw the answer given...  

I am to find the second derivative of:
x(x^2+1)^2

I found the first as:
4x^2(x^2+1)+(x^2+1)^2

Is this correct?  If not, how do I simplify this further?  

I try to continue on but I notice the book drops the square at the end on the (x^2+1)^2
and I am not sure why which just begins to confuse me.  

This is from the chapter on higher derivatives in business calculus.  Any help would be great!

ANSWER: Questioner:   Aaron
Category:  Calculus
Private:  No
 
Subject:  Need help with finding second derivative
Question:  I am confused when it comes to finding the second derivative of a function.  An example of this is with the function:

f(x) = (x^2+2)^5

I find the first derivative as
10x(x^2+2)^4
Then, in looking at the equation I figure that I need to use the product rule next.

I take the first:
(10x)
times the derivative of the second
(4)(x^2)^3 (2x)
and add that to the second:
(x^2+2)^4
times the derivative of the first:
(10)

This gives me:
(10x)(4)(x^2+2)^3(2x)+(x^2+2)^4(10)

I am unsure of how to proceed next.  The solutions manual says that the second derivative answer is:

10(9x^2+2)(x^2+2)^3??

The other problem I am working on is very similar and I felt confident until I saw the answer given...  

I am to find the second derivative of:
x(x^2+1)^2

I found the first as:
4x^2(x^2+1)+(x^2+1)^2

Is this correct?  If not, how do I simplify this further?  

I try to continue on but I notice the book drops the square at the end on the (x^2+1)^2
and I am not sure why which just begins to confuse me.  

This is from the chapter on higher derivatives in business calculus.  Any help would be great!
 
.....................................................
If you have

f' = 10x(x^2+2)^4

Then, using the product rule:

f'' = (10x)(8x(x^2 + 2)^3) + (10)(x^2 + 2)^4

Next factor out  10(x^2 + 2)^3  -- the highest C.F.

f'' =  (10(x^2 + 2)^3)( x(8x) + (x^2 + 2))

f'' =  (10(x^2 + 2)^3)( 8x^2 + x^2 + 2)

f'' =  (10(x^2 + 2)^3)( 9x^2 + 2)

So you are Ok; you just have to remember to do this factoring.
.................................
For your second one:

I found the first as:
4x^2(x^2+1)+(x^2+1)^2

Looks like the same thing -- factor out  (x^2 + 1)

(x^2+1)[ 4x^2 + x^2 + 1 ]

(x^2+1)[ 5x^2 + 1 ]

I think that's about it.


---------- FOLLOW-UP ----------

QUESTION: I am confused about when and what you need to look for to begin factoring.  I see and understand why you used the product rule to begin the second derivative on the first problem, but then I am not sure where the part you factor out comes from.

On the second problem, it is the same thing.  It looks like I understand the basic derivatives, but not what to look for to factor.  Also, on the first problem, could you not multiply the 10x and 8x by each other to get 80x^2 at that point?

Answer
Hi, again, Aaron,

QUESTION: I am confused about when

>> Whenever you have more than one term.  [And after 10:00 A.M. -- math is best done later in the morning.]

and what you need to look for to begin factoring.  

>> Common factors, of course.  When you see:

10x(8x(x^2 + 2)^3)

you will say:

That has a factor of 2, and 5, and 10, and 8, and x, and
three factors of (x^2 + 3)

And when you see:

(10)(x^2 + 2)^4

you say:

It has a factor of 10 and four factors of (x^2 + 2).  

Then you will say:

I see a common factor of 10 and three C.F.'s of (x^2 + 2)

and go on from there
.......................
I see and understand why you used the product rule to begin the second derivative on the first problem, but then I am not sure where the part you factor out comes from.

On the second problem, it is the same thing.  It looks like I understand the basic derivatives, but not what to look for to factor.  Also, on the first problem, could you not multiply the 10x and 8x by each other to get 80x^2 at that point?

BITE YOUR TONGUE!!!  Don't say things like that.  You are trying to factor, and multiplying out is the opposite thing.

[Sorry for the intemperate language.]