  The bounded derived category of a finite dimensional algebra of finite global
dimension is equivalent the stable category of $\mathbb{Z}$-graded modules over
its trivial extension \cite{Happel}. In particular, given two derived
equivalent finite dimensional algebras $\Lambda_{1}$ and $\Lambda_{2}$ of
finite global dimension, their trivial extensions are stable equivalent. The
converse is not true in general. The goal of this paper is to study cases where
derived equivalences between $\Lambda_{1}$ and $\Lambda_{2}$ arise from an
equivalence of categories involving their trivial extension. Thanks to a graded
version of Happel's theorem, we show that one can construct a
$\mathbb{Z}$-grading on the $\Lambda_{i}$ so that such an equivalence involving
their trivial extension yield a derived equivalence between the category of
$\mathbb{Z}$-graded module over $\Lambda_{i}$. We describe explicitly the
tilting object associated to this derived equivalence (in the non-graded and in
the graded case) for triangular matrix algebras. Finally, we apply these
results to the particular case of gentle algebras. In this context, we study
how one can obtain a derived equivalence between $\Lambda_{1}$ and
$\Lambda_{2}$ (in the non-graded and the graded case) from graded generalized
Kauer moves.
