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Find derivative of an integral with limits

  1. Apr 12, 2013 #1
    I'm in analysis and I'm trying to understand the following.

    1. The problem statement, all variables and given/known data

    g(x) = integral from 0 to x+δ of f(x)dx + integral from x-δ to 0 of f(x)dx

    g'(x) = f(x+δ) - f(x -δ)

    So how do they get g'(x)?
  2. jcsd
  3. Apr 12, 2013 #2


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    By Leibnitz rule for differentiation of an integral as a function of the upper limit.
    By the way, it's better to use a dummy variable in the integrand:$$
    \frac d {dx}\int_a^x f(t)\, dt = f(x)$$
  4. Apr 12, 2013 #3
    So I understand why there isn't an f(a) since the derivative of a constant is zero, but like in my problem one of my limits is zero and since the function isn't given it could be anything, even something like f(x) = 1/x which at zero is undefined, but in my problem it just goes away to zero. Why? I hope you understand what I'm trying to say.
  5. Apr 12, 2013 #4


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    There are hypotheses on Leibnitz's rule. You can't even talk about, for example, things like$$
    F(x) = \int_0^x \frac 1 t\, dt$$because the integral is divergent.
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