Goodwillie calculus – approximation of homotopy theories by stable homotopy theories
In (∞,1)-categorification of the chain rule in ordinary differential calculus, in Goodwillie calculus the Goodwillie chain rule is a formula for describing the Goodwillie derivatives of a composite functor $F \circ G$ in terms of the Goodwillie derivatives of $F$ and $G$.
John R. Klein, John Rognes, A chain rule in the calculus of homotopy functors, Geom. Topol. 6 (2002) 853-887 (arXiv:math/0301079)
Jacob Lurie, section 6.3 of Higher Algebra
Gregory Arone, Michael Ching, Operads And Chain Rules For The Calculus Of Functors, arXiv:0902.0399
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