  We show that the leading derivative corrections to the Heisenberg-Euler
effective action can be determined efficiently from the vacuum polarization
tensor evaluated in a homogeneous constant background field. After deriving the
explicit parameter-integral representation for the leading derivative
corrections in generic electromagnetic fields at one loop, we specialize to the
cases of magnetic- and electric-like field configurations characterized by the
vanishing of one of the secular invariants of the electromagnetic field. In
these cases, closed-form results and the associated all-orders weak- and
strong-field expansions can be worked out. One immediate application is the
leading derivative correction to the renowned Schwinger-formula describing the
decay of the quantum vacuum via electron-positron pair production in
slowly-varying electric fields.
