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Differentiating an Integral

  1. Jun 24, 2007 #1
    Hi, everyone, I'm new here and don't know how to type mathematics, but I have a scanner.


    I have a function L_A and it is an integral. I want to differentiate this function with respect to A. I already have the answer written but what I don't know is how it was obtained.

    Just by looking at the answer I can sort of see some sort of pattern, and I have written what I think is some sort of rule on the second half of this page, but I still don't really know what kind of differentiation rule is used here, so if any smart people here know it would greatly help me thanks!
  2. jcsd
  3. Jun 24, 2007 #2


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    If you first perform the whole integration, then differentiating. I.e. get an explicit expression for L_A.
    What do you get then? Have you tried that?
  4. Jun 24, 2007 #3


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    In general, Leibniz's rule, an extension of the fundamental theorem of calculus, says:
    [tex]\frac{\partial left(\int_{f(a,s)}^{g(a,s)} \phi(a,s,x)dx\right)}{\partial a}= \frac{\partial g}{\partial a}\phi(a,s,f(a,s))- \frac{\partial f}{\partial a}\phi(a,s,g(a,s))+ \int_{f(a,s)}^{g(a,s)} \frac{\partial \phi(a,s,x)}{da} dx[/tex]
    just the form you give. It can be derived using the fundamental theorem of calculus together with the chain rule to handle the variable limits of integration.
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