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

Almost global solutions to two classes of 1-d Hamiltonian Derivative Nonlinear Schr\"odinger equations