  Regarding the wide applications of dilaton gravity in the presence of
electrodynamics, we introduce a suitable Lagrangian for the coupling of dilaton
with gauge field. There are various Lagrangians which show the coupling between
scalar fields and electrodynamics with correct special situations. In this
paper, taking into account conformal transformation of Brans-Dicke theory with
an electrodynamics Lagrangian, we show that how scalar field should couple with
electrodynamics in dilaton gravity. In other words, in order to introduce a
correct Lagrangian of dilaton gravity, one should check at least two
requirements: compatibility with Brans-Dicke theory and appropriate special
situations. Finally, we apply the mentioned method to obtain analytical
solutions of dilaton-Born-Infeld and Brans-Dicke-Born-Infeld theories with
energy dependent spacetime.
