  We give a derivative version of the relative trace formula on PGL(2) studied
in our previous work, and obtain a formula of an average of central values
(derivatives) of automorphic $L$-functions for Hilbert cusp forms. As an
application, we prove existence of Hilbert cusp forms with non-vanishing
central values (derivatives) such that the absolute degrees of their Hecke
fields are sufficiently large.
