Find an integral expression for $\Gamma'(z)$ for $\Re z\gt 0$

Question

Find an integral expression for $\Gamma'(z)$ for $\Re z\gt 0$

I know the result. But I dont know how to show this step by step.

Answer 1

Using Leibniz's rule for differentiation under the integral sign (why can we?):

$$\Gamma'(z)=\frac{d}{dz}\int\limits_0^\infty x^{z-1}e^{-x}dx=\int\limits_0^\infty x^{z-1}e^{-x}\log x\,dx$$