So we have \Gamma(s)=\int_0^\infty t^{s-1} e^{-t} ~dt=\int_0^\infty f(s,t) ~dt, s \in \mathbb{R}^{>0}
In order to say that \Gamma'(s)=\int_0^\infty \frac{\partial f}{\partial s} (s,t) ~dt=\int_0^\infty t^{s-1} \ln(t) e^{-t} ~dt, we have to prove, in particular, that there exists an integrable function g such that :
\left|\frac{\partial f}{\partial s}(s,t)\right|\leqslant g(t)

My idea is as following :
For any n, there exists a>1 such that e^t>t^n \Rightarrow e^{-t}<t^{-n}
For any t>a, \ln(t)\leqslant t
So we have :
\left|\frac{\partial f}{\partial s}(s,t)\right|\leqslant \left|t^{s-1} \cdot t \cdot t^{-n}\right|

By letting n=s+2, we have :
\left|\frac{\partial f}{\partial s}(s,t)\right|\leqslant \frac{1}{t^2}, which is integrable over [a,+\infty)

So I wanted to know if this part is correct... especially the red sentence :s

And for the integral over 0 and \epsilon, I don't know how to dominate

Thanks in advance ^^

i really hate these problems, especially if it's for helping friends xD