My favorite example of why τ is the true circle constant is actually the equation A = 1/2 τ r^2. It fits the usual quadratic form and shows how circumference and area relate: c = dA/dr = τ r and so A = ∫ c dr = ∫ τ r dr = 1/2 τ r^2. At first glance those 1/2s seem superfluous and like it would be nicer to just have a 2 in the other terms, but the 1/2 shows that differentiation and integration with respect to r are happening, where the "1/2" and the "^2" anhialate each other on differentiation and then re-form on integration.
Reminds me of Taylor series. The factorial terms eluded me for so long until I realized it was a cancelling factor for accumulated derivation of polynomials. Now there's this link in my mind between powers / derivation / factorials.
What polynomials are being derived, and from what premises? Are you thinking of how to derive series solutions to differential equations or something? I'm unclear on what you're getting at here.
(Or, oh: did you mean "differentiation" when you said "derivation"? That would fit what you've said. Sorry for the pedantic post; I'll just leave this here in case anyone else is confused.)
Sorry for the fuzzy vocabulary. TBH I wasn't even convinced I understood taylor series correctly. A quick search on wikipedia hint at that derivation is used sometimes; derivative; differentiation etc etc.
About the point:
d(n)(x^n.dx)
= n d(n-1)(x^(n-1).dx)
= (n * (n-1 * (...)))
= n! * x
hence the 1/n! term.
