1. 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!

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 $g(t) = \int_{t}^{\infty} f(T) \, dT = - \int_{\infty}^{t} f(T) \, dT$.

3. 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?