To prove that $\displaystyle \frac {d}{dt} \int^{b}_{a} f(x,t) \ dx = \int^{b}_{a} \frac{d}{dt} f(x,t) \ dx $ at $\displaystyle t=t_{0} $

is it sufficient to show that $\displaystyle f(x,t)$ and $\displaystyle \frac{d}{dt}f(x,t)$ are continous for $\displaystyle x$ in the range of integration and for $\displaystyle t$ in a interval about $\displaystyle t_{0} $? And if you want to differentiate under the integral sign multiple times, is it sufficient to show that each derivative of $\displaystyle f(x,y) $ is continous for $\displaystyle x$ in the range of intergration and for $\displaystyle t $ in a interval about $\displaystyle t_{0}$?

And what about improper integral $\displaystyle \frac {d}{dt} \lim_{b \to \infty} \int^{b}_{a} f(x,t) \ dx $ ?

Is it sufficient to show in addition that $\displaystyle |f(x,t)| $ and $\displaystyle \Big|\frac{d}{dt}f(x,t)\Big|$ are bounded above by functions independent of $\displaystyle t$ and the integrals of both of those functions converge over the interval [a,$\displaystyle \infty) $?