A chain rule in the calculus of homotopy functors

We formulate and prove a chain rule for the derivative, in the sense of Good-willie, of compositions of weak homotopy functors from simplicial sets to simplicial sets. The derivative spectrum partial derivative F(X) of such a functor F at a simplicial set X can be equipped with a right action by the loop group of its domain X, and a free left action by the loop group of its codomain Y = F(X). The derivative spectrum partial derivative(E o F)( X) of a composite of such functors is then stably equivalent to the balanced smash product of the derivatives partial derivative E(Y) and partial derivative F(X), with respect to the two actions of the loop group of Y. As an application we provide a non-manifold computation of the derivative of the functor F(X) - Q(Map(K, X)+).