  Let $A$ be a finite dimensional associative algebra with derivations over a
field of characteristic zero, i.e., an algebra whose structure is enriched by
the action of a Lie algebra $L$ by derivations, and let $c_n^L(A),$ $n\geq 1,$
be its differential codimension sequence. Such sequence is exponentially
bounded and $\exp^L(A) = \lim_{n\to \infty}\sqrt[n]{c_n^L(A)}$ is an integer
that can be computed, called differential PI-exponent of $A$.
  In this paper we prove that for any Lie algebra $L$, $\exp^L(A)$ coincides
with $\exp(A)$, the ordinary PI-exponent of $A$. Furthermore, in case $L$ is a
solvable Lie algebra, we apply such result to classify varieties of
$L$-algebras of almost polynomial growth, i.e., varieties of exponential growth
such that any proper subvariety has polynomial growth.
