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2nd derivative trouble

  1. Aug 25, 2008 #1
    1. The problem statement, all variables and given/known data

    Find horiz. asymptotes and use them w/ concavity and intervals to sketch the curve

    2. Relevant equations

    3. The attempt at a solutionf'(x)= 1-x^2/(x^2+1)^2
    But then I can't seem to work through taking the 2nd derivative, perhaps I am not using the chain rule right.
    I get -4x^5-2x^3-2x/(x^2+1)^4
    But thats not right... please help!
  2. jcsd
  3. Aug 25, 2008 #2


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    Assuming you meant f(x)=x/(x^2+1) and not f(x)=(x/x^2)+1, the first derivative is correct. So again using the quotient rule, the second derivative will be
    [tex]\frac{ (1 - x^2)' (x^2 + 1)^2 - (1 - x^2) ((x^2 + 1)^2 )' }{(x^2+1)^4}[/tex]
    Can you work out the two derivatives that are there separately?
  4. Aug 25, 2008 #3
    So I get that f''(x)= ((-2x)((x^2+1)^2)-1-x^2)((2)(x^2+1))(2x))/(x^2+1)^4
    I think thats right but not sure about distributing through.

    I get (2x^5-2x^3-4x)/(x^2+1)^4

    But the posted answer is (2x^5-4x^3-6x)/(x^2+1)^4
  5. Aug 25, 2008 #4


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    The first line is correct, so you probably did something wrong in the expansion.
    Try working out
    separately, and only then adding them.
    Forgetting about the denominator (x^2 + 1)^4 for a while, you should get 2 x^5 - 4x^3 - 6x.

    And look at the bright side: probably it's some stupid writing error, at least you know you can differentiate :smile:
  6. Aug 25, 2008 #5
    Thanks for all the help, most of my errors are in the algebra...
    on the left side I get (-2x^5-2x) and the right -((1-x^2)(4x^3+4x)) which goes to -(4x^3+4x-4x^5-4x^3).... Left - Right I get 2x^5-6x!! still missing the -4x^3 I don't know where i am going wrong.
  7. Aug 26, 2008 #6


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    You do? I don't.
    What does (x^2 + 1)^2 expand to? It's not x^4 + 1!
    If you want, write it out: (a + b)^2 = (a + b)(a + b) = ....
    Then remember that formula (or at least, remember to remember that something is going on whenever you see it) forever :smile:
  8. Aug 26, 2008 #7
    now I understand....I was convinced my mistake was on the right not the left. Thanks a ton.
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