Differential calculus for fully matricial functions with emphasis on cyclic derivatives

We will introduce appropriate cyclic derivatives for fully (stably) matricial functions and study their basic properties. In particular, we will prove a Poincaré lemma for stably matricial functions of certain classes. We will also position Voiculescu's framework of fully matricial functions in the context of nc functions due to Kaliuzhnyi-Verbovetskyi and Vinnikov.