Gauge covariance of the charged Dirac equation ($D_{\mu }^{\prime }{\Psi }^{\prime }=e^{-iq\alpha }{D}_{\mu }\Psi$)

The derivative of the local phase in $\Psi \mapsto e^{-iq\alpha }\Psi$ cancels the added gradient in the electromagnetic potential. Thus $D_{\mu }^{\prime }{\Psi }^{\prime }=e^{-iq\alpha }{D}_{\mu }\Psi$, and the charged Dirac equation maps solutions to solutions. The same cancellation makes its action gauge invariant.