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We prove a chain rule for the Goodwillie calculus of functors from spectra tospectra. We show that the (higher) derivatives of a composite functor $FG$ at abase object $X$ are given by taking the composition product (in the sense ofsymmetric sequences) of the derivatives of $F$ at $G(X)$ with the derivativesof $G$ at $X$. We also consider the question of finding $P_n(FG)$, and give anexplicit formula for this when $F$ is homogeneous.

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