Differentiation of integrals over time dependent regions

Question

Is there a general equation relating $$ \int_{\Omega(t)} \frac{\partial f}{\partial t} dx $$ and $$ \frac{d}{dt} \int_{\Omega(t)} f dx $$ If $\Omega$ is constant and $f$ is smooth (edit: and bounded), then these are known to be equal. How, if at all, does the time dependence of $\Omega$ change the relationship of these two integrals?