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