  Our aim in this paper is to investigate the first Hochschild cohomology of
{\em admissible algebras} which can be seen as a generalization of basic
algebras. For this purpose, we study differential operators on an admissible
algebra. Firstly, differential operators from a path algebra to its quotient
algebra as an admissible algebra are discussed. Based on this discussion, the
first cohomology with admissible algebras as coefficient modules is
characterized, including their dimension formula. Besides, for planar quivers,
the $k$-linear bases of the first cohomology of acyclic complete monomial
algebras and acyclic truncated quiver algebras are constructed over the field
$k$ of characteristic $0$.
