Recall that given a $k$-algebra $A$, a derivation on $A$ is a $k$-linear morphism $d:A\to A$ such that $$d(ab)=d(a)b+ad(b).$$

The use of derivations is of paramount importance in mathematics. I think that it would be interesting to collect different definitions that are given the name of derivation or might share similarities with the classical definition and maybe comment on their usefulness or why they were created. They can come from any branch of mathematics.