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Math Help - Difficult Differentiation

  1. #1
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    Difficult Differentiation

    Taking the derivative of  \int ^ { \infty } _{t} e^ {- \int ^{v} _{t} r ( \omega ) d \omega }

    Where v and t represents the same variable (in this case, time) at different value, with v being fixed and t being the variable of differentiation.

    I know that I will probably have to use Leibniz's rule at some point, but I'm threw off by the infinity sign, how do I take the derivative with that? (Can I still use the fundamental theorem of calculus?)

    Thanks!
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Taking the derivative of  \int ^ { \infty } _{t} e^ {- \int ^{v} _{t} r ( \omega ) d \omega }

    Where v and t represents the same variable (in this case, time) at different value, with v being fixed and t being the variable of differentiation.

    I know that I will probably have to use Leibniz's rule at some point, but I'm threw off by the infinity sign, how do I take the derivative with that? (Can I still use the fundamental theorem of calculus?)

    Thanks!
    I've got several issues with your notation.

    Should it not be something like  \int ^ { \infty } _{t} e^ {- \int ^{v} _{{\color{red}T}} r ( \omega ) d \omega } \, {\color{red}dT} .... ?

    In which case you have something of the form g(t) = \int_{t}^{\infty} f(T) \, dT = - \int_{\infty}^{t} f(T) \, dT.
    Last edited by mr fantastic; September 21st 2008 at 04:36 PM. Reason: Added the red for easier comparison
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    I believe so, this is the free entry condition for technological research cost in Macro Economic.

    So I can still use the fundamental theorem of calculus here?
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  4. #4
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    Quote Originally Posted by tttcomrader View Post
    I believe so, this is the free entry condition for technological research cost in Macro Economic.

    So I can still use the fundamental theorem of calculus here?
    Yes.
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