  In this paper we study the growth of the differential identities of some
algebras with derivations, i.e., associative algebras where a Lie algebra $L$
(and its universal enveloping algebra $U(L)$) acts on them by derivations. In
particular, we study in detail the differential identities and the cocharacter
sequences of some algebras whose sequence of differential codimensions has
polynomial growth. Moreover, we shall give a complete description of the
differential identities of the algebra $UT_2$ of $2\times 2$ upper triangular
matrices endowed with all possible action of a Lie algebra by derivations.
Finally, we present the structure of the differential identities of the
infinite dimensional Grassmann $G$ with respect to the action of a finite
dimensional Lie algebra $L$ of inner derivations.
