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The aim of this paper is three-fold: (i) we construct a naturally occurringhighly homotopy coherent operad structure on the derivatives of the identityfunctor on structured ring spectra which can be described as algebras over anoperad $\mathcal{O}$ in spectra, (ii) we prove that every connected$\mathcal{O}$-algebra has a naturally occurring left action of the derivativesof the identity, and (iii) we show that there is a naturally occurring weakequivalence of highly homotopy coherent operads between the derivatives of theidentity on $\mathcal{O}$-algebras and the operad $\mathcal{O}$.  Along the way, we introduce the notion of $\mathbf{N}$-colored operads withlevels which -- by construction -- provides a precise algebraic framework forworking with and comparing highly homotopy coherent operads, operads, and theiralgebras.

On the Goodwillie derivatives of the identity in structured ring spectra