Derived plethystic algebra

We prove comparison results about $\mathrm{\infty }$-categories of derived commutative rings with extra algebraic structure. For example, we show that several notions of derived $\delta$-rings agree and obtain similar results for derived $\lambda$-rings, derived binomial rings, and derived Boolean rings. Our comparison results apply more generally to modules over a specific kind of monad on derived rings that we call derived plethory.