Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 30E / b / ii / Solution

After $t=zs$, the phase in the integral for $\Gamma'(z)$ becomes $\log s-s$, with maximum $-1$ and second derivative $-1$ at $s=1$. Hence
$$
\Gamma'(z)\sim\sqrt{\frac{2\pi}{z}}e^{z\log z-z}\log z.
$$


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 Incoming links: Laplace asymptotic for the derivative of the Gamma function