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: http://www.math.mcgill.ca/rags/difftl/cartdiff-tac.pdf http://www.math.mcgill.ca/rags/difftl/faaslide.pdf https://www.mta.ca/uploadedFiles/Community/Bios/Geoff_Cruttwell/sman3.pdf http://www.tac.mta.ca/tac/volumes/32/9/32-09.pdf
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)
