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Find the second derivative of F (x)

  1. Dec 7, 2009 #1
    1. The problem statement, all variables and given/known data

    Assuming sufficient differentiability, find second derivative of F(x) = integ[a,x] (t-x)2 f(t) d(t)
    2. Relevant equations

    Probably Fund.Thm of Calculus and some properties

    3. The attempt at a solution

    I really have no idea..I tried evaluating but with t=x but I get zero..I have never done second derivative so I'm somewhat clueless. Anyone care to help and explain?
  2. jcsd
  3. Dec 7, 2009 #2


    Staff: Mentor

    The second derivative is the derivative of the first derivative. E.g., if f(x) = x^2, f'(x) = 2x, and f''(x) = d/dx(f'(x)) = d/dx(2x) = 2.

    The Fundamental Theorem of Calculus can help you get F'(x), so take a careful look at this theorem and any examples that show how to use it. After that, just take the derivative of what you got for F'(x).
  4. Dec 7, 2009 #3


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    Use the Leibniz integral rule twice. Besides the term that you found equal to zero, there's another term that involves differentiating the integrand. That part is not necessarily zero.
  5. Dec 7, 2009 #4
    I don't quite understand? I've never learned the leibniz rule. Can you exemplify it with this question? The f(t) part also confuses me.
    Last edited: Dec 7, 2009
  6. Dec 7, 2009 #5


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    Don't worry about the f(t) part. Your answer will be in terms of f(t). http://en.wikipedia.org/wiki/Leibniz_integral_rule Here's an easy example. Take F(x) to be the integral from a to x of x*t*dt. You only have to deal with the first and third terms in the rule. The first term you get just from putting t=x in the integrand, so x^2. That's the one you know. The last term says you should differentiate inside the integral. That gives you the integral from a to x of t*dt. Or x^2/2-t^2/2. Adding the two you get F'(x)=3*x^2/2-a^2/2. Compare that with what you get by actually integrating and then differentiating. Now try doing that with your example. What's F'(x)?
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