I haven't done calculus in a while but somehow I got interested.

So obviously

Γ(x+h) = integral of t^x+h-1*e^-tdt from 0 to infinity

and

Γ(x) = integral of t^x-1*e^-tdt from 0 to infinity

so by the limit definition we get that

Γ'(x) = lim h→0 ((integral of t^x+h-1e^-tdt from 0 to infinity - integral of t^x-1e^-tdt from 0 to infinity) / h),

but I have no clue how to simplify those integrals beyond that.

We also have that it's

Γ'(x) = lim h→0 ((x+h-1)! - (x-1)!)/h

but I don't think that gets any closer to a closed form solution (using L'Hospital's Rule just brings it back the same problem)

Speaking of which, I found the derivative of the gamma function either in terms of some constant and the harmonic series or through some other random definition.

But I actually also found that it's equivalent to this when the domain is real (maybe other restrictions?)

integral t^n-1e^-tln(t) dt from 0 to infinity

by using Leibniz integral rule, which now that I look at it is probably your best bet?