Axiomatic approach to differential calculus?
This is called "differential algebra". Google the term.
Sometimes the product rule (or "Leibniz rule") is stated as follows: $(fg)' = f'g + fg'$, with $f$ on the left and $g$ on the right in every term. In that way, this can be applied to noncommutative rings.
Related to Michael Hardy's answer, you could look at the concepts of derivations and of Kähler differentials in commutative algebra. These are used to formalize the concepts of differential forms and exterior derivatives in the context of algebraic geometry over fields other than $\mathbb R$ of $\mathbb C$.