The product rule is defined for operator valued functions, and will then also be defined for functions whose values are in Banach algebras. It al depends on the definition of the derivative. The following definition I find particularly useful

where is then the derivative of at . The situation can be simplified slightly when we consider functions of the form

with a Banach algebra. The above definition then reduces to

.

Here the resemblance with the normal definition of the derivative is clear. So let us now consider the product rule

.

Just check my algebra, but I think this should clear it up.