= Solution
A useful differentiation theorem says: if $f(t,x)$ is integrable in $x$, differentiable in $t$ for almost every $x$, and on a <neighborhood> of each parameter value the derivative is <bounded> in absolute value by one <integrable function> of $x$, then the integral is differentiable and its derivative is the integral of $\partial_tf$. The proof applies <dominated convergence> to difference quotients, <bounded> by that same majorant by the <mean value theorem>.
Fix $t_0>0$ and choose $0<a<t_0<b$. For $t\in[a,b]$,
$$
|e^{-x}x^{t-1}\log x|\leq\begin{cases}x^{a-1}|\log x|,&0<x\leq1,\\ e^{-x}x^{b-1}\log x,&x\geq1.\end{cases}
$$
The first part is integrable by part (i). The second is integrable by the permitted hypothesis for $h_b$; alternatively <exponential decay> dominates its <polynomial> and logarithmic factors. The theorem therefore applies locally at every $t_0$, giving
$$
\boxed{\Gamma'(t)=\int_0^\infty e^{-x}x^{t-1}\log x\,dx\qquad(t>0).}
$$
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