= Solution
Let $U(x)=e^{ie\alpha(x)}$, $\psi'=U\psi$, and $A'_\mu=A_\mu+\partial_\mu\alpha$. Direct differentiation in the <gauge covariant derivative> gives
$$
D'_\mu\psi'=(\partial_\mu-ieA'_\mu)(U\psi)
=U\left[\partial_\mu+ie\partial_\mu\alpha-ieA_\mu-ie\partial_\mu\alpha\right]\psi
=UD_\mu\psi.
$$
The gradient of the local phase is exactly cancelled by the potential transformation. Since $U$ is scalar in spinor space and commutes with the <Dirac matrices>,
$$
\boxed{(i\gamma^\mu D'_\mu-m)\psi'=U(i\gamma^\mu D_\mu-m)\psi.}
$$
Consequently a solution is mapped into a solution: the minimally coupled <Dirac equation> is covariant under the local <U(1) gauge symmetry>. The operator and wavefunction both transform; ordinary differentiation of the transformed wavefunction alone would leave an extra phase-gradient term.
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