Derivative of differential form

Question

A smooth differential form $\omega$ on a smooth manifold $M$ is a map $\omega: M \to \bigwedge M$. Since $\bigwedge M$ is a vector bundle, and hence a manifold, if $\omega$ is differentiable as a map $M \to \bigwedge M$ we can form the tangent map $$T\omega: TM \to T\left( \bigwedge M \right).$$ Is a smooth differential form always differentiable in this sense and is there a relationship to the exterior derivative of the differential form? If not, what is the importance of this derivative?