As usual when differentiating something with respect to a variable that appears twice.  The chain rule for partial derivatives.

For example, consider function $z = f(u,v)$.  Suppose we want $(d/dt)f(t,t)$.  Let $u=v=t$ and use
 $dz/dt = (\partial z/\partial u)(du/dt) + (\partial z/\partial v)(dv/dt)$.

Thus...
$$
  \frac{d}{dt}\int_0^t\int_0^t f(x,y)\\,dx\\,dy = 
  \int_0^t f(t,y)\\,dy + \int_0^t f(x,t)\\,dx
$$

By the way, why did you write $\partial/\partial t$ to differentiate a function of the single variable $t$?  It's not wrong, just confusing to students.