Two-region domination for differentiation of the gamma integral ($|{\partial }_t(e^{-x}{x}^{t-1})|\le x^{a-1}|\log x|1_{x\le 1}+e^{-x}{x}^{b-1}\log x{1}_{x>1}$)

For $t$ in a compact positive interval $[a,b]$, split the integration region at one. Near zero the smallest exponent gives the displayed bound, whose integral is $1/a^2$. Above one the largest exponent gives the bound, integrable by exponential decay. A single integrable function therefore dominates all parameter derivatives locally. Dominated differentiation under the integral sign gives ${\Gamma }^{\prime }(t)={\int }_0^{\infty }{e}^{-x}{x}^{t-1}\log xdx$. The same argument with higher powers of $|\log x|$ gives all higher derivatives.