  We study the structure possessed by the Goodwillie derivatives of a pointed
homotopy functor of based topological spaces. These derivatives naturally form
a bimodule over the operad consisting of the derivatives of the identity
functor. We then use these bimodule structures to give a chain rule for higher
derivatives in the calculus of functors, extending that of Klein and Rognes.
This chain rule expresses the derivatives of FG as a derived composition
product of the derivatives of F and G over the derivatives of the identity.
  There are two main ingredients in our proofs. Firstly, we construct new
models for the Goodwillie derivatives of functors of spectra. These models
allow for natural composition maps that yield operad and module structures.
Then, we use a cosimplicial cobar construction to transfer this structure to
functors of topological spaces. A form of Koszul duality for operads of spectra
plays a key role in this.
