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Please help. Points of inflection and chain rule

  1. Nov 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Skethch the greaph of x^3/(x^3+1). Identify all extrema and points of inflection, asymptote equations, and easily found intercepts

    2. Relevant equations

    If a/b=0, a must be 0.(thats how I got critical points from first derivative)
    And chain rule: F'(x) = f '(g(x)) g '(x)
    And so on.

    3. The attempt at a solution

    What I have to do is to find points of inflection, asymptotes, and critical points. And I took the first derivative using Quotient rule and got 3x^2/(x^3+1)^2
    I am going to take the second derivative of my original function to get ponts of inflection, but I am stuck.

    As you can see, f'(x) = 3x^2/(x^3+1)^2
    And I pondered a bit for I think I need to use chain rule because of that (x^3+1)^2 in the denominator. But how should I use it? what about the numerator?
    I can't just use chain rule in the denominator, and use power rule in nominator on my own, right? I have a bad feeling about this.

    This is what I will get if I apply power rule in nominator and apply chain rule in denominator:

    6x/[2(x^3+1)(3x^2)] This is wrong, right? because I treated numerator and denominator by using different rules on each of them. Can you check this for me? What should I do in order to get the second derivative?
    Or can I just ignore the denominator and just take Quotient rule again to get derivative?
     
  2. jcsd
  3. Nov 11, 2009 #2
    Your second derivative doesn't look quite right.
    Since you're having trouble getting the second derivative, try doing it in steps, starting from f'(x) = 3x2/(x3 + 1)2
    Let g(x) = 3x2 and h(x) = (x3 + 1)2. Find g' and h' and then just plug everything into the right side of the quotient rule formula
    [tex]\left(\frac{g}{h}\right)' = \frac{h'g - gh'}{h^2}[/tex]
    and simplify.
     
  4. Nov 12, 2009 #3
    Now I get it. I was quite chaotic when I first saw this problem. Thank you for your advice!
     
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