What is ∂∂xi(xi!) where xi is a discrete variable?
Do you consider (xi!)=(xi)(xi−1)...1 and do product rule on each term, or something else?
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+n∑k=11k) where γ is the Euler-Mascheroni constant.