Derivative of a factorial

What is xi(xi!) where xi is a discrete variable?

Do you consider (xi!)=(xi)(xi1)...1 and do product rule on each term, or something else?
Thanks.

Answer

The derivative of a function of a discrete variable doesn’t really make sense in the typical calculus setting. However, there is a continuous variant of the factorial function called the Gamma function, for which you can take derivatives and evaluate the derivative at integer values.

In particular, since n!=Γ(n+1), there is a nice formula for Γ at integer values:
Γ(n+1)=n!(γ+nk=11k) where γ is the Euler-Mascheroni constant.

Attribution
Source : Link , Question Author : Grizzly , Answer Author : icurays1

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