I wondered in which cases, given a measurable space $(A,\mu)$, Banach spaces $E,F$, an open $U\subseteq E$ and $f:A\times U\rightarrow F$, we can conclude that the function $s\mapsto\int_A f(x,y)dx$ is smooth on $U$. (Obviously we need some smoothness and integrability assumptions on $f$ but I couldn't figure out exactly which. Possibly $A$ should also be assumed a topological space.)
Another thing which came to my mind is that I don't know if it is enough for proving continuous differentiability of a function between Banach spaces to prove the existence and continuity of all directional derivatives (as it is the case in finite dimensional spaces).
Also I would be happy about all recommendations of books / online resources that deal with differentiation in Banach spaces, thanks!