I'm trying to understand how Wolfram_alpha reached this equation:
So far i reached: $$ \frac{\partial\Gamma(a,x)}{\partial a} = \frac{\partial}{\partial a}\int_x^\infty t^{a-1}e^{-t}dt = \int_x^\infty t^{a-1}e^{-t}ln(t)dt $$
in which $\ln(t)$ is the natural logarithm of $t$. According to Wolfram_alpha it can be rewritten as:
$$ \frac{\partial\Gamma(a,x)}{\partial a} = \Gamma(a)^2z^a \,_2F_2(a,a;a+1,a+1;-x)-\Gamma(a,0,x)ln(x)+\Gamma(a)\psi(a) $$
Source: http://functions.wolfram.com/GammaBetaErf/Gamma2/20/01/01/0002/