Digamma duplication?

Legendre duplication formula for gamma function gives that:

$\begin{aligned} \Gamma (2x) & = \frac {2^{2x-1}}{\sqrt \pi} \Gamma (x) \Gamma \left(x + \frac 12\right) & \small \color{#3D99F6} \text{Take natural logarithm both sides} \\ \ln \Gamma (2x) & = \ln \left(2^{2x-1}\right) - \frac 12 \ln \pi + \ln \Gamma (x) + \ln \Gamma \left(x + \frac 12\right) & \small \color{#3D99F6} \text{Differentiate both sides w.r.t }x \\ 2 \frac d{dx} \ln \Gamma (2x) & = \frac {2^x \ln 2}{2^{2x-1}} + \frac d{dx} \ln \Gamma (x) + \frac d{dx} \ln \Gamma \left(x + \frac 12\right) & \small \color{#3D99F6} \text{Note that }\psi (x) = \frac d{dx} \ln \Gamma (x) \\ 2 \psi (2x) & =2\ln 2 + \psi (x) + \psi \left(x + \frac 12\right) \end{aligned}$

$\begin{aligned} \implies \exp \left(2 \psi (2\pi) - \psi (\pi) - \psi \left(\pi + \frac 12\right) \right) & = \exp \left(\ln 4 \right) = \boxed{4} \end{aligned}$