It is easy to show that the differential forms of order $1$ obeys a form of chain rule. To be precise, $d(f(g(x)) = f^\prime(x) d(g(x))$. This can be for example proved by fixing a co-ordinate basis $x_i$ for the tangent space at some point, and looking at the action of $df$ on each $\partial/\partial x_i$.
My first question is whether there exists a co-ordinate free proof of this.
The more important question is whether some such rule exists for higher order differential forms. The motivation is my attempt to grasp how similar differential forms and the usual process of differentiation are.