# Derivative of a factorial

What is ${\partial\over \partial x_i}(x_i !)$ where $x_i$ is a discrete variable?

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

In particular, since $n!=\Gamma(n+1)$, there is a nice formula for $\Gamma^\prime$ at integer values:
where $\gamma$ is the Euler-Mascheroni constant.