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Disclaimer, I am a physicist and mess up with math and really think that derivatives are just fractions (roughly).

I am starting to study maths itself as the discipline it is and not as a tool for my science. Concretely, I am delving into differential forms and manifolds because there are some facts about physics that I do not really understand. For example, why is there a Lie algebra underlying Poisson Brackets, why the phase space is a 2-form and its role in geometric quantization.

In this journey, an intuition is growing (that I do not know if is correct) on what is a derivative, let me explain. I have seen that every operator that somehow follows a Leibniz rule can be understood as a derivative. The derivation applied to functions, the external derivative and the Poisson Bracket follows a Leibniz rule-like. Regarding the last one, as I have seen, in the free coordinate formulation of differential forms and manifolds the Poisson Brackets are very familiar to Lie Derivatives.

This motivated my question: is the Leibniz rule fundamental in the descriptions of derivatives? I mean, some operator that do not follow this kind of rule implies that is not a derivative?

Thanks in advance for your response.

T.

T. ssP
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    Relevant rabbit hole to dive into: https://en.wikipedia.org/wiki/Derivation_(differential_algebra) – Eero Hakavuori Nov 22 '23 at 08:38
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    Very, very fundamental, one can see it as what actually defines derivations. Some people (I am one of them) prefer to define the extensions to the whole tensor algebra of Lie derivatives, covariant derivatives, etc. precisely thanks to Leibniz rule. – Didier Nov 22 '23 at 08:41
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    That very recent question may be handy: https://math.stackexchange.com/questions/4806434/soft-question-generalizations-of-the-derivative/ "Derivative" may be understood as something defined by Leibniz rule (and linearity wrt addition and scalar multiplication), but it can also be generalized, with variants among Leibniz rules. Finite differences (https://en.wikipedia.org/wiki/Finite_difference, https://en.wikipedia.org/wiki/Divided_differences) should be mentionned too, they also show Leibniz-like rules. – Jean-Armand Moroni Nov 22 '23 at 08:45

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