  In this paper we investigate in details derivations on trivial extension
algebras. We obtain generalizations of both known results on derivations on
triangular matrix algebras and a known result on first cohomology group of
trivial extension algebras. As a consequence we get the characterization of
trivial extension algebras on which every derivation is inner. We show that,
under some conditions, a trivial extension algebra on which every derivation is
inner has necessarily a triangular matrix representation. The paper starts with
detailed study (with examples) of the relation between the trivial extension
algebras and the triangular matrix algebras.
