What can be said about the relation between the domain and range of a derivation as a function?
If $A$ is the domain, any space of functions, what does $D(A)$ look like, where $D$ is a derivation? Is $D(A)=A$? I guess the function space should always be closed under multiplication and addition and then the Leibniz rule
$$D(ab)=D(a)b+aD(b)$$
should be unproblematic. It comes down to the definition of $D$ acting on the functions. In case that the domain is a priori "not as big" as the range, can one complete a function space by adding the closure w.r.t. a derivative?
And are there any categorical results? After all the cohomology business deals with mappings given by differential operators.