How to derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n+ 1)$? How can you derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n + 1)$? I have tried checking Wolfram Alpha for a step-by-step solution, but none is given. Moreover, of what is the second function, $\psi^{(0)}$, representative?
$\Gamma(n)=(n-1)!$

$\Gamma(n+1)=n!$

${d\over{dn}}n!={d\over{dn}}\Gamma(n+1)$ 

now just look arround you for the derivative of the gamma function
$\Gamma(n)=(n-1)!$

$\Gamma(n+1)=n!$

${d\over{dn}}n!={d\over{dn}}\Gamma(n+1)$ 

now just look arround you for the derivative of the gamma function