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


And also many other solution sets to classes of differential equations. You can also replace $x$ by any elementary function in these classes. 

