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.