Let $ A $ be a Poisson algebra, $ M $ be a left Poisson module over $ A $. Then, there is a Poisson algebra structure on the trivial extension algebra $A\ltimes M$ of $ A $ by $ M $. Assuming that $ M $ is $ A $ itself or the linear dual $ {A}^{*} $of $ A $, then both $A\ltimes {A}^{*}$ and $A\ltimes A$ are Frobenius Poisson algebras. The modular derivation for these two classes of the trivial extension of Poisson algebras was calculated. These results can be viewed as the corresponding conclusions on the Nakayama automorphism of the trivial extension of finite dimensional algebras.