In some sense the title is a bit misleading (one is led to think of Analytic Number Theory). I'd rather use the title "Using differentials to...", which is more precise, as there is not exactly any "calculus" going on but there is indeed differential algebra and differential "number theory", so to speak.

But great and elegant article. Thanks.

The technical term is "formal derivative" b/c there are no limits involved, it's basically a rewrite rule for changing x^n to nx^(n-1).