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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.

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    $\begingroup$ This seems to be a good question for ChatGPT, possibly with some critical postprocessing and subsequent multistage search. $\endgroup$
    – Iosif Pinelis
    Feb 5 at 19:19
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    $\begingroup$ nLab gives a quite extensive list, anything missing? ncatlab.org/nlab/show/derivation $\endgroup$
    – Carlo Beenakker
    Feb 5 at 19:23
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    $\begingroup$ For example, $p$-derivations and arithmetic derivatives (of course $p$-derivation is in the related concepts of the nlab page). And maybe there are some derivation-like notions that are not (at least conceptually) straightforward generalizations of the algebraic derivation. $\endgroup$
    – Arturo
    Feb 5 at 19:44
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    $\begingroup$ Possibly relevant are the Mathematics Stack Exchange questions Is there a garden of derivatives? (doesn't deal with algebraic and number-theoretic analogues, however) and Calculus and Category theory. $\endgroup$
    – Dave L Renfro
    Feb 5 at 23:38

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