Expert: Paul Klarreich - 10/16/2006

Question
I'm having difficulty find the f''(x) of f(x) = (x^2)/[(x^2)+3]

I know the f'(x) = 6x[(x^2-1)]^2

I'm getting confused if I need to apply the product rule for this step
of the problem.

In the end I need to get this second derivative into a format that
shows me all the critical numbers such as what value of x will result
in the y value equaling 0.  Thanks.

Answer
john rhem Asks in Category Calculus ...
 
Subject:  second derivatives
 
Question:  I'm having difficulty finding the f''(x) of f(x) = (x^2)/[(x^2)+3]

I know that f'(x) = 6x[(x^2-1)]^2

>> do you indeed?

I'm getting confused if I need to apply the product rule for this step of the problem.

>> The short answer to that is, YES.

In the end I have to get this second derivative into a format that shows me all the critical numbers such as what value of x will result in the y value equaling 0.  

>> Do you mean the y' = 0?  y'' = 0?

Thanks.
...........................................
Hi, John,

There seems to be a glitch in your calculations.  You write that

        x^2  
f(x) = --------
      x^2 + 3  

Is that correct?  If so, you should be getting your derivative like this:

        x^2  
f(x) = ---------- = 1 - 3(x^2 + 3)^-1,  by long division.
      x^2 + 3  

f'(x) = 3(x^2 + 3)^(-2)(2x)
      6x
= -------------
  (x^2 + 3)^2

Or, if you use the quotient rule:

       (x^2 + 3)(2x) - (x^2)(2x)
f'(x) = -------------------------
           (x^2 + 3)^2


        2x^3 + 6x - 2x^3
f'(x) = ------------------ = same as before.
           (x^2 + 3)^2

...............................................
Now, if you have:

           6x
f'(x) = -----------
       (x^2 + 3)^2

The answer to your question is, unfortunately, yes, you have to use the quotient rule.  HOWEVER, if you are looking for critical points, when you are done, all you have to do is set the NUMERATOR of your answer equal to zero.  

NOTE: When you set y'' = 0 and solve, what you get are the INFLECTION POINTS.  You get critical points from setting y'=0.

Let me know how it comes out.