The arithmetic derivative (discussed here: https://plus.google.com/101584889282878921052/posts/9nY35Ma1pbU) is cute but perhaps not that important.
However differentiable functors seem important for programming. They take data structures and return a "one hole context" for that structure. This makes it possible to write code that traverses data structures in a very general way.
In http://stackoverflow.com/questions/25554062/zipper-comonads-generically/25572148#25572148, Connor McBride shows how this can involve partial differentiation. Wow.
As we see it is pretty messy in Haskell, even with multifarious optional features enabled. I suspect that this is stuff which is an indicator of whether you've got your programming language design right.