Derivative of the gamma function (${\Gamma }^{\prime }(a)={\int }_0^{\infty }{e}^{-t}{t}^{a-1}\log tdt$)

For positive real $a$, differentiating the Gamma function integral under the integral sign gives the displayed formula; successive derivatives insert higher powers of $\log t$. Dominated differentiation on compact positive-$a$ intervals is justified by exponential decay at infinity and integrability of $t^{a-1}|\log t{|}^n$ at zero. At $a=1$, these derivatives give moments used in logarithmic endpoint asymptotic expansions.