Infinity-operads and Day convolution in Goodwillie calculus

We prove two theorems about Goodwillie calculus and use those theorems to describe new models for Goodwillie derivatives of functors between pointed compactly-generated infinity-categories. The first theorem say that the construction of higher derivatives for spectrum-valued functors is a Day convolution of copies of the first derivative construction. The second theorem says that the derivatives of any functor can be realized as natural transformation objects for derivatives of spectrum-valued functors. Together these results allow us to construct an infinity-operad that models the derivatives of the identity functor on any pointed compactly-generated infinity-category. Our main example is the infinity-category of algebras over a stable infinity-operad, in which case we show that the derivatives of the identity essentially recover the same infinity-operad, making precise a well-known slogan in Goodwillie calculus. We also describe a bimodule structure on the derivatives of an arbitrary functor, over the infinity-operads given by the derivatives of the identity on the source and target, and we conjecture a chain rule that generalizes previous work of Arone and the author in the case of functors of pointed spaces and spectra.