  Consider two kinds of 1-d Hamiltonian Derivative Nonlinear Schr\"odinger
(DNLS) equations with respect to different symplectic forms under periodic
boundary conditions. The nonlinearities of these equations depend not only on
$(x,\psi,\bar{\psi})$ but also on $(\psi_x,\bar{\psi}_x)$, which means the
nonlinearities of these equations are unbounded. Suppose that the
nonlinearities depend on the space-variable $x$ periodically.
  Under some assumptions, for most potentials of these two kinds of Hamiltonian
DNLS equations, if the initial value is smaller than $\varepsilon\ll1$ in
$p$-Sobolev norm, then the corresponding solution to these equations is also
smaller than $2\varepsilon$ during a time interval
$(-c\varepsilon^{-r_*},c\varepsilon^{-r_*})$(for any given positive $r_*$). The
main methods are constructing Birkhoff normal forms to two kinds of Hamiltonian
systems which have unbounded nonlinearities and using the special symmetry of
the unbounded nonlinearities of Hamiltonian functions to obtain a long time
estimate of the solution in $p$-Sobolev norm.
