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 \( c_n^L(A), n\ge 1 \), be its differential codimension sequence. Such sequence is exponentially bounded and \( \mathrm{exp}^L(A)=\lim_{n\to\infty}\sqrt[n]{c_n^L(A)} \) is an integer that can be computed, called differential PI-exponent of \( A \). In this paper we prove that for any Lie algebra \( L \), \( \mathrm{exp}^L(A) \) coincides with \( \mathrm{exp}(A) \), 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.