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There are various differentiations/derivatives.

For example,

  • Exterior derivative $df$ of a smooth function $f:M\to \mathbb{R}$
  • Differentiation $Tf:TM\to TN$ of a smooth function between manifolds $f:M\to N$
  • Radon-Nikodym derivative $\frac{d\nu}{d\mu}$ of a $\sigma$-finite measure $\nu$
  • Fréchet derivative $Df$ of a function between Banach spaces $f:V\to W$

What is the most general definition of differentiation or derivative?

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    $\begingroup$ In a different direction there is the notion of approximate differentiaility for maps defined on not necessarily open but merely measurable subsets of Euclidean spaces. $\endgroup$ – Behnam Esmayli Dec 4 '19 at 13:04
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    $\begingroup$ @Behnam Esmayli: I don't understand your comment, because even the ordinary derivative can easily be defined for functions defined on an arbitrary set at a limit point of the set (i.e. given a set $E$ and a point $x \in E,$ consider the ordinary limit of the ordinary difference quotient for $x$ and points in $E$ that approach $x).$ That said, I'll mention that there are a lot of variations on the notation of an approximate derivative both for the ordinary difference quotient and for variations on the ordinary difference quotient (see here). $\endgroup$ – Dave L Renfro Dec 4 '19 at 18:19
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    $\begingroup$ I think the idea that there is one most general definition is probably wrong. $\endgroup$ – LSpice Dec 4 '19 at 18:49
  • $\begingroup$ The appx derivative I have in kind exists at density points. One important fact about them is that Lipschitz maps from a measurable subset $E$ have appx derivative a.e. and that area formula holds using this derivative's Jacobian in the formula. There is also the more general Stepanov's theorem. I just wanted to point to this direction in looking for a more general notion. $\endgroup$ – Behnam Esmayli Dec 4 '19 at 23:39
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That's a real can of worms. There are tons of different notions of differentiability for functions lacking classical smoothness: the Gateaux derivative, the weak derivative, the distributional derivative, the directional derivative, the subgradient (for convex functions), Clarke's generalized gradient, Hadamard differentiability, Bouligand differentiability, the metric derivative and the upper gradient to name a few. All these notions can't be ordered by "generality" in any way I am aware of. And there are even notions of differentiability for set-valued maps...

I think the reason for the large number of notions is that differentiability has distinct motivations: local approximation by a simpler structure (e.g. by linear maps (add some sort of continuity if you like)), measuring the rate of change (add a notion of direction if you like), inverting the process of integration in some sense, finding descent directions for optimization, an algebraic ruleset for polynomials…

Oh, and you may want to have a look at this answer which contains 12(!) more notions of differentiability.

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    $\begingroup$ @DanieleTampieri: according to ijpam.eu/contents/2013-83-3/7/7.pdf, given $X, Y$ normed vector spaces, a map $f:X\to Y$ is B-differentiable at $x$ if $\exists L$ such that $\| f(y) - f(x) \| \leq L \|y - x\|$ for every $y$ in a small open neighborhood of $x$, and for every vector $v$, the limit $\lim_{t \searrow 0} \frac{1}{t} (f(x + tv) - f(x) )$ exists. $\endgroup$ – Willie Wong Dec 4 '19 at 15:07
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    $\begingroup$ @DanieleTampieri: essentially the difference between Bouligand and Frechet (as described in jstor.org/stable/pdf/44001767.pdf) is that in Frechet differentiability, you assume the function is at $x$ approximated by a linear function, while in Bouligand you assume the function is at $x$ approximated by a function that is merely one-homogeneous. $\endgroup$ – Willie Wong Dec 4 '19 at 15:09
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    $\begingroup$ In some cases, some notion is stronger than another, but in other cases they are not comparable (e.g. the weak derivative is used for functions on real domains and the subgradient is defined for convex functions on normed spaces and even in the cases where both are applicable they do not really compare well). $\endgroup$ – Dirk Dec 4 '19 at 17:41
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    $\begingroup$ How about a partial order? $\endgroup$ – user76284 Dec 5 '19 at 0:39
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    $\begingroup$ To complement the "real functions of a real variable" variations in "this answer" of mine you cited, and somewhat different from the various multivariable (including infinite-dimensional) variations you mentioned, there is a seemingly endless variety of notions based on work done in the 1930s through 1960s (mostly) on derivation bases that grew out of attempts to refine and generalize the Fundamental Theorem of Calculus by Busemann/Feller, Zygmund, Saks, Denjoy, Trjitzinsky, Haupt, Pauc, Morse, Hayes, Guzman, and others. (continued) $\endgroup$ – Dave L Renfro Dec 5 '19 at 10:33
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I guess that depends on what you mean by most general, and what qualifies as a derivative. There are some purely syntactic definitions of differentiation that come up in category theory.

Cartesian differential categories axiomatize a differentiation operator that satisfies all of the higher order chain rules from normal differential calculus (and any differentiation operator that satisfies those higher order chain rules will give you a Cartesian differential category due to a free construction of Cockett and Seely).

Tangent categories axiomatize the differentiation function of maps between manifolds. They can be described as categories with an action by the category of Weil algebras, which satisfies the same properties as Weil prolongation in the category of smooth manifolds.

I’m writing this on my phone so I’ll just post links at the bottom here:

  • R.F. Blute, J.R.B. Cockett and R.A.G. Seely, Cartesian differential categories, Theory and Applications of Categories, Vol. 22, 2009, No. 23, pp 622-672. (abstract)
  • J.R.B. Cockett and R.A.G. Seely, The Faà di Bruno construction, Theory and Applications of Categories, Vol. 25, 2011, No. 15, pp 393-425. (abstract) (slides from a talk by Seely)
  • J. R. B. Cockett and G. S. H. Cruttwell, Differential Structure, Tangent Structure, and SDG, Applied Categorical Structures 22 (2014) 331–417. doi:10.1007/s10485-013-9312-0, (author pdf)

  • Poon Leung, Classifying tangent structures using Weil algebras, Theory and Applications of Categories, Vol. 32, No. 9, 2017, pp. 286–337 (abstract)

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In addition to the generalizations mentioned in other answers, there is an abstract version of differentiation featured in differential algebras. This operator retains the familiar differentiation rules for sums and products of operands, but is otherwise agnostic as to how the operation is obtained. Differential algebra is applied particularly in nonlinear control theory.

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Other answers possibly did not mention multiplicative differentiation (inverse of multiplicative integral) and discrete differentiation (finding of finite difference, both backward and forward). There is also discrete multiplicative differentiation. I do not think there is any generalization that includes these ones.

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