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Eventhough the concept of the derivative is always the same, there exist different ways of stating it and generalising it.

My question is: What different definitions of the derviative exist and in which context do they make sense?

For example, one of the definitions of the derivative is the Fréchet derivative, which is defined on Banach spaces.

groupoid
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One rather axiomatic approach is that you call any operator $D$ a derivative that is

  1. Linear:

    • $D(\alpha f) = \alpha Df$
    • $D(f+g) = Df+Dg$

  2. Obeys Leibniz' Rule:

    • $D(fg)=fDg + gDf$

For "scalars" $\alpha$ and "functions" $f$, $g$. This allows to define derivatives even for functions that operate on discrete sets$^1$ where no concepts like limes or completeness are available. All you need is "addition" and "multiplication", both commutative$^2$.

That is the scalars are $\def\M{\mathcal S} \alpha\in \M=(\M,+,\cdot)$, functions are $f\in \{\M\to\M\}$, and functionals are $D\in \{\{\M\to\M\} \to \{\M\to\M\}\}$

$^1$Not to be confused with discrete derivative.

$^2$Where the latter condition could be dropped. In that case you'll come up with different flavours.

emacs drives me nuts
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    Cool! Do you know who came up with this definition? – groupoid Mar 11 '20 at 18:08
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    No. But good overview is https://en.wikipedia.org/wiki/Generalizations_of_the_derivative for different approaches to derivative. There are also axiomatic approaches to Lie derivative or Exterior derivative, just to name two. – emacs drives me nuts Mar 12 '20 at 11:06