  The derivative expansion of the one-loop effective Lagrangian in QED$_4$ is
considered. The first term in such an expansion is the famous Schwinger result
for a constant electromagnetic field. In this paper we give an explicit
expression for the next term containing two derivatives of the field strength
$F_{\mu\nu}$. The results are presented for both fermion and scalar
electrodynamics. Some possible applications of an inhomogeneous external field
are pointed out.
