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