I thought up this method to find $\frac{d}{ds}\Gamma (s)$, and I want to know if it is valid. I know that there are other questions about this on MSE but they don't really help me understand whether or not what I do is valid.
Given: $$\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}dt$$ Thus: $$\Gamma'(s)=\lim_{h\to 0}\frac{1}{h}\Biggl(\int_0^\infty t^{s+h-1}e^{-t}dt-\int_0^\infty t^{s-1}e^{-t}dt\Biggr)$$ $$\therefore \Gamma'(s)=\lim_{h\to0}\frac{1}{h}\int_0^\infty \Biggl(t^{s+h-1-1}e^{-t}-t^{s-1}e^{-t}\Biggr)dt$$ $$\therefore \Gamma'(s)=\int_0^\infty t^{s-1}e^{-t}\biggl(\lim_{h\to0}\frac{t^h-1}{h}\biggr)dt$$ $$\therefore \Gamma'(s)=\int_0^\infty t^{s-1}e^{-t}\ln t\;dt$$ Right?
Yes, your proof is correct. You have demonstrated every step of the proof clearly.