I just knocked up a quick document showing the relation of the classical definition of the derivative and the limits imposed by Special Theory of Relativity (the "time dilation"/ action creation diagram).
(The actual definition of Newton's using "fluxions" applies to our everyday macro world, and this analysis shows the relevance of Bohr's Correspondence Principle - the morphing to continuity from discrete action in the limit h -> 0)
Relativistic Limits (updated)
(Note; x' doesn't have to actually go to infinity; just enough so that a (i.e. "h") is insignificant when x' is compared to x. This corresponds to an ultraviolet cutoff...
I will be adding to an revising this document occasionally, so this is just a start..