TITLE: Most general definition of differentiation
QUESTION [23 upvotes]: 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?

REPLY [2 votes]: 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.