I've been trying to evaluate how $\partial_{\mu}\phi$ transforms under *infinitesimal* Lorentz transformations. If $\partial_{\mu}\phi \mapsto (\partial_{\mu}\phi)'=(\Lambda^{-1})^{\nu}_{}_{\mu}\partial_{\nu}\phi((\Lambda^{-1})^{\nu}_{}_{\mu} x^{\mu})$, then for $\Lambda^{\mu}_{}_{\nu}=\delta^{\mu}_{\nu}+\omega^{\mu}_{}_{\nu}$, how does this expression simplify?
One thing that I tried was $$\partial_{\mu}\phi(x^{\mu}) \mapsto \frac{\partial \phi(\bar{x}^{\mu})}{\partial \bar{x}^{\mu}}=\frac{\partial x^{\mu}}{\partial \bar{x}^{\mu}}\frac{\partial \phi(x^{\mu}+\omega^{\mu}_{}_{\nu}x^{\nu})}…