  We study the Higgs derivative interactions on models including arbitrary
number of the Higgs doublets. These interactions are generated by two ways. One
is higher order corrections of composite Higgs models, and the other is
integrating out heavy scalars and vectors. In the latter case, three point
couplings between the Higgs doublets and them are the sources of the derivative
interactions. The representations of these heavy particles are constrained to
couple with the doublets. We explicitly calculate the all derivative
interactions generated by integrating out. Their degrees of freedom and
conditions to impose the custodial symmetry are discussed. We also study the
vector boson scattering processes with a couple of two Higgs doublet models to
see experimental signals of the derivative interactions. They are differently
affected by each heavy field.
