  Interactions between derivatives and fixpoints have many important
applications in both computer science and mathematics. In this paper, we
provide a categorical framework to combine fixpoints with derivatives by
studying Cartesian differential categories with a fixpoint operator. We
introduce an additional axiom relating the derivative of a fixpoint with the
fixpoint of the derivative. We show how the standard examples of Cartesian
differential categories where we can compute fixpoints provide canonical models
of this notion. We also consider when the fixpoint operator is a Conway
operator, or when the underlying category is closed. As an application, we show
how this framework is a suitable setting to formalize the Newton-Raphson
optimization for fast approximation of fixpoints and extend it to higher order
languages.
