In this note we consider homomorphisms between differentiable
Lipschitz algebras $Lip^n(X,\alpha)$ ($0<\alpha \leq 1$) and
$lip^n(X,\alpha)$ ($0<\alpha <1$), where $X$ is a perfect compact
plane set. We give sufficient conditions implying the compactness
and power compactness of these homomorphisms. Moreover, we investigate
under what conditions a quasicompact homomorphism between these algebras is power
compact. We also give a necessary condition for a homomorphism
between these algebras to be quasicompact and in certain cases to
be power compact. Finally, using these results, by giving an
example we show that there exists a quasicompact homomorphism
between these algebras which is not power compact.