This document has two purposes, one is to establish the product rule and quotient rule from calculus not by using the “add-zero trick” that is typically displayed. We establish a relationship for the expression that will aid in the proof of these results. Since the “add-zero trick” is not the easiest for students to apply, this result provides an alternative and purely arithmetic way of proving these rules. The second purpose is to generalize the product rule and the quotient rule. We shall generalize the product rule to a formula that can be applied with M functions as factors while considering the nth order derivative of such a product. Finally, we shall construct a quotient rule for derivatives of order n.

product rule, quotient rule, calculus, Leibniz rule, Faà di Bruno formula.