A chain rule for Goodwillie derivatives of functors from spectra to spectra

Abstract

We prove a chain rule for the Goodwillie calculus of functors from spectra to spectra. We show that the (higher) derivatives of a composite functor F G at a base object X are given by taking the composition product (in the sense of symmetric sequences) of the derivatives of F at G(X) with the derivatives of G at X. We also consider the question of finding Pn(F G), and give an explicit formula for this when F is homogeneous.