Deformations of $\mathbb{E}_{\infty}$-groups of units and logarithmic derivatives of $\mathbb{E}_{\infty}$-rings

Abstract: We extend a classical fact about deformations of groups of units of commutative rings to $\mathbb{E}_{\infty}$-ring spectra, and we use this result to provide a map of spectra generalizing the ordinary logarithmic derivative induced by an $R$-module derivation.