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Math Help - Differentiating intergral of composite function

  1. #1
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    Differentiating intergral of composite function

    Hi I'm looking for help with a simple question in calculus. The question is: how do you differentiate the integral of a composite function? Specifically, if what is ?

    The closest thing to an answer I could find was Leibniz's rule, which suggested the answer should be , but the answer I should get (according to someone else's notes) is just , as if the integral isn't there.

    I actually think there might not be an answer, i.e. there may be some hand-waving going on. I would be grateful if someone could confirm if that's the case. If there is a rule that applies to this situation, I would be grateful if someone could explain.

    Thanks.
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  2. #2
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    Re: Differentiating intergral of composite function

    Just in case a picture helps...



    ... where (key in spoiler) ...

    Spoiler:


    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case j), and the straight dashed line similarly but with respect to the dashed balloon expression (k, the inner function of the composite which is subject to the chain rule).

    Full size

    So, I don't think Liebniz' rule applies. But what is the context of the question?


    _________________________________________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
    Last edited by tom@ballooncalculus; November 13th 2012 at 12:39 PM.
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  3. #3
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    Re: Differentiating intergral of composite function

    Hi,

    Thanks for your help. The diagram does clarify things somewhat. I had already thought of using the chain rule, but I realized that since I don't know \frac{dk}{dj} it wasn't much help. I think you're right about Leibniz's rule: I was looking for a rule that applied to differentiating integrals and that was the best I could find. It seemed more plausible because the original problems concern multivariable functions (see below) so I overlooked the fact that the key here was composite functions. Plus, someone told me I was right to use Leibniz's rule, but I think he may have been wrong.

    The question is actually based on two problems I have encountered. I just focused on the key issue to avoid complicating things, and because I don't really know LaTex so typing the equations is a pain. Anyway, the main one concerns a function of the form Y(t)=AL^\alpha H(t)^\beta \int k(j,t)^{1- \alpha - \beta} \, dj and this is supposed to give \frac{\partial Y(t)}{\partial k(j,t)}=(1-\alpha-\beta)AL^\alpha H(t)^\beta k(j,t)^{-\alpha -\beta}. The idea is that Y(t) is output at time t and k(j,t) are amounts of production inputs (basically machines) of type j at each point in time. So the derivative is supposed to be the rate of change of output with respect to a particular type of input.

    The other problem concerns the Lagrangian L=\int_0^\infty \left(e^{-\rho t}u(c(t)+\lambda(t) \left(f(k(t)-c(t)-\frac{dk(t)}{dt}\right)\right) dt where the first-order condition is supposed to be \frac{\partial L}{\partial c(t)}=e^{-\rho t}u'(c(t))-\lambda(t)=0.

    So, in both cases it's like you can just ignore the integral and just differentiate the function within. I'm just trying to figure out what the mathematical justification for that is (assuming there is one!).
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