Classic American bowler (5)
I believe the answer is:
This clue is a double definition.
'classic' is the first definition.
(Derby City Classic pool tournament)
'american bowler' is the second definition.
(I've seen this before)
(Other definitions for derby that I've seen before include "Flat race; hat", "Type of hat", "English city", "Epsom 20", "Annual race at Epsom".)