  We show that the gauge-invariant kinetic equation of superconductivity
provides an efficient approach to study the electromagnetic response of the
gapless Nambu-Goldstone and gapful Higgs modes on an equal footing. We prove
that the Fock energy in the kinetic equation is equivalent to the generalized
Ward's identity. Hence, the gauge invariance directly leads to the charge
conservation. Both linear and second-order responses are investigated. The
linear response of the Higgs mode vanishes in the long-wave limit. Whereas the
linear response of the Nambu-Goldstone mode interacts with the long-range
Coulomb interaction, causing the original gapless spectrum lifted up to the
plasma frequency as a result of the Anderson-Higgs mechanism, in consistency
with the previous works. The second-order response exhibits interesting
physics. On one hand, a finite second-order response of the Higgs mode is
obtained in the long-wave limit. We reveal that this response, which has been
experimentally observed, is attributed solely to the drive effect rather than
the widely considered Anderson-pump effect. On the other hand, the second-order
response of the Nambu-Goldstone mode, free from the influence of the long-range
Coulomb interaction and hence the Anderson-Higgs mechanism, is predicted. We
find that both Anderson-pump and drive effects play important role in this
response. A tentative scheme to detect this second-order response is proposed.
