This is a bit of a trivial question, but as I don't know the answer immediately I thought I'd just ask.

Given the integral $\int_{0}^{t} \int_{0}^{t} f(x,x') dx dx'$, what is $\frac{\partial}{\partial t} \int_{0}^{t} \int_{0}^{t} f(x,x') dx dx'$? It looks a bit like differentiating under the integral sign, but I'm not sure how to handle it.

notlike differentiating under the integral sign; moreover, without some conditions on $f$ it's not clear to me that you can say anything precise) – Yemon Choi Apr 1 '10 at 18:25`$f$`

continuous, just draw the darn square and ask yourself what the difference is between integrating over`$[0,t]^2$`

and over`$[0,t+{\rm d}t]^2$`

. – Theo Johnson-Freyd Apr 1 '10 at 21:01