Thus, to Lorentz transform the $\psi$ part of $\psi(x)$ at $x$ without changing $x$ we send $\psi(x) \to \psi'(x) = \Lambda_{1/2} \psi(x_0) = \Lambda_{1/2} \psi(\Lambda^{-1}x)$. Now, if we Lorentz transform a differential equation, like the Dirac equation, because $\partial_{\mu}$ explicitly depends on $x$ we must also Lorentz transform it.