The Image of Derivations on Banach Algebras of Differential Functions

Let $D:C^n(I){\longrightarrow}M$ be a derivation from the Banach algebra of n times continuously differentiable functions on an interval I into a Banach $C^n(I)$-module M. If D is continuous and D(z) is contained in the k-differential subspace, the image of D is contained in the k-differential subspace. The question of when the image of a derivation is contained in the k-differential subspace is discussed.