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Determining the derivative of an integral

  1. Nov 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Given: the integral from 0 to infinity of t^(x)e^(-t)dt
    Problem: Determine f'(x).


    2. Relevant equations



    3. The attempt at a solution
    My teacher mentioned using the definition of a derivative:
    f'(a)= limit as x approaches a of f(x)-f(a)/(x-a).

    So far I have: f'(a)=the integral of the limit as x approaches a of t^(x)e^(-t)-t^(a)e^(-t)/(x-a) dt.

    I'm not sure where to go from here, or if thats even correct. I think it should end up being f'(x)=t^(x)e^(-t) which makes sense to me. I'm just not sure how to actually get there.

    Thank you to all.
     
  2. jcsd
  3. Nov 27, 2009 #2

    Dick

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    Look at the Leibniz rule for differentiation of an integral. If you actually want to do the resulting integral, you'll need to use a gamma function, but maybe you can just give the answer in terms of another integral.
     
  4. Nov 28, 2009 #3

    ideasrule

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    I don't quite understand this. What's the x in t^(x)e^(-t)dt? Is it just a constant?

    Anyhow, the integral from 0 to infinity of that function should be a constant, like 2 or 2.313. Deriving a constant gives you 0.
     
  5. Nov 28, 2009 #4

    Dick

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    Changing the value of x changes the value of the integral. Try x=0, x=1, etc. It's not a constant.
     
  6. Nov 28, 2009 #5

    ideasrule

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    Oh. Sorry, I misunderstood the question.
     
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