Is there a finite set $P$ of non-elementary functions $f\_n$ such that the derivative of any function $f$ from that set is not elementary, but expressible with functions from the same set $P$ plus elementary functions?



Is there a class of functions closed against differentiation besides elementary?
Let $g$ be an elementary function whose indefinite integral $f$ is nonelementary, and set $P = \{\cos(t)f(t), \sin(t)f(t)\}$.


