  In this paper, we consider associative algebras equipped with derivations.
Such a pair of an associative algebra with a derivation is called an AssDer
pair. Using the Hochschild cohomology for associative algebras, we define
cohomology for an AssDer pair with coefficients in a representation. We study
central extensions and abelian extensions of AssDer pairs. Moreover, we
consider extensions of a pair of derivations in central extensions of
associative algebras. Next, we study formal one-parameter deformations of
AssDer pair by deforming both the associative product and the derivation. They
are governed by the cohomology of the AssDer pair with representation in
itself. In the next part, we study $2$-term $A_\infty$-algebras with homotopy
derivations considered by Loday and Doubek-Lada. Finally, we introduce
$2$-derivations on associative $2$-algebras and show that the category of
associative $2$-algebras with $2$-derivations are equivalent to the category of
$2$-term $A_\infty$-algebras with homotopy derivations.
