Is there a garden of derivatives? I've found a book called A Garden of Integrals, in which the author shows the evolution of the concept of Integral. I follow AnalysisFact on Twitter, some days ago, they posted the following:

The got curious about the generalizations of the derivatives. I'm wondering if there's something analog to the book I mentioned: Some kind of garden of derivatives.
 A: Without even leaving the realm of real valued functions of a real variable, I can think of: approximate derivative, preponderant derivative, path derivative, symmetric derivative, approximate symmetric derivative, pseudosymmetric derivative, qualitative derivative, strong derivative, Peano derivative, first return derivative, and there are others obtainable by combining these, such as symmetric qualitative derivative and first return path derivative, not to mention one-sided versions, Dini derivate versions, upper (lim sup) two-sided versions, lower (lim inf) two-sided versions [note that the Dini derivate versions are the upper and lower one-sided versions], etc.
See also these books:
Differentiation of Real Functions by Andrew M. Bruckner
Real Functions by Brian S. Thomson
Theory of Differentiation by Krishna M. Garg
Higher Order Derivatives by Satya Mukhopadhyay
(ADDED 5.5 YEARS LATER) Since this has gotten a lot of attention lately and my original answer was very quickly made (it was originally a comment that I converted to an answer), it would probably be a good idea to include the following additional remarks, which were made recently as comments in mathoverflow.
To complement the "real functions of a real variable" variations, there are also 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. A very nice survey of this work is given in Andrew M. Bruckner's Differentiation of Integrals and the more recent survey paper Differentiation by Brian S. Thomson (Chapter 5, pp. 179-247, in Vol. I of Handbook of Measure Theory). And to really get a sense of how deep this rabbit hole goes, see Derivation and Martingales by Hayes/Pauc, Differentiation of Integrals in R$^n$ by Guzman, and the infamous Web Derivatives by Kenyon/Morse. For why I say "infamous" regarding this last publication, see the last paragraph (under "P.S.") of this 25 October 2000 sci.math post. Regarding those sci.math comments, see also Joachim Lambek's review of Morse's 1965 book A Theory of Sets on pp. 354-355 of Canadian Mathematical Bulletin 11 #2 (June 1968).
