ca.analysis and odes - Differentiate under integral sign for iterated integral?

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...
$$
frac{d}{dt}int_0^tint_0^t f(x,y)\,dx\,dy =
int_0^t f(t,y)\,dy + int_0^t f(x,t)\,dx
$$

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.