We can define a derivation over a field $K$ to be a map $D:K\rightarrow K$ such that
$D(u+v) = Du + Dv$ and $D(uv) = uDv + vDu$,
for $u,v\in K$, and the constant subfield
$Const_D(K) = \lbrace u\in K : Du=0 \rbrace$
of $K$ with respect to $D$.
From this, we can get the familiar analytic properties of differentiation.
My question is: are there any other operators similar to this which we can define over a field, from which we then get some interesting relation to the analytic world.