Yes, I did, but I didn't fully grasp it. Anyway, this is what I can come up with, please take a look and let me know if it makes (physical) sense.
Definition: q(r,z,t)=δ(t)Q(r,z)
\frac{\partial q(r,z,t)}{\partial t} = Q(r,z) \frac{d}{dt}[δ(t)]
\frac{\partial q(r,z,t)}{\partial t} = Q(r,z)...