In this paper, we present a generic differentiation rule that is applicable to all mathematical operators and illustrate how the generic differentiation rule is vital in deducing the derivatives of complex mathematical operations such as functional iteration. We also show how the generic rule offers more insight into the concept of differentiation and provides systematic proofs to differentiation rules that, otherwise, would have been proven using ad hoc approaches. Next, the generic rule is shown to yield interesting results including a proof that the first-degree approximation of the composite of iterated functions, at the limit where the iteration variable approaches zero, is the sum of the iterators of those functions. Consequently, at the vicinity of the specified limit, composition of iterated functions is remarkably symmetric.
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