Probably the Fréchet derivative is what is wanted.
Please, tell me how I can prove this, seemingly, simple statement?
If - the Banach spaces, - an open set and mappings are differentiable at the point , then the mappings and also differentiable at the point , and with equalities
P.S. How do you define differentiability in Banach spaces?
Probably the Fréchet derivative is what is wanted.