Update: This was one of my early adventures into mathematics, and
it's been a long time since I've done anything more on it,
but over the years others reading this have brought to my attention
various similar pursuits, so I'll mention them here for the benefit
of those who maintain an interest in it:
Prof. John Verosky of Western Washington University
developed something similar to product calculus
referred to as a "biological calculus" (discussions in 1999).
Michael Doré briefly
derived similar results for product calculus. Others have mentioned
similar ideas (Casey Coleman and Shane Taylor).
For more details, search for "generalized calculus david"
on the messageboard http://math2.org/mmb/ .
Surely yet others have also considered replacing addition and
multiplication operators in Newtonian calulus with different operators,
as has been done with algebra.
Thanks to Bob Brown
for pointing out the Dannon paper and other discussions.