Coarse person in Derby, a hooligan (5)
I believe the answer is:
yahoo
'coarse person' is the definition.
(a yahoo is a rude or noisy person)
'in derby a hooligan' is the wordplay.
'in' indicates the answer is hidden within the clue.
'YAHOO' is hidden within 'derby a hooligan'.
(Other definitions for yahoo that I've seen before include "Swiftian brute" , "Brutish animal species from Gulliver's Travels" , "Brute, lout" , "Swift chap" , "A Swiftian lout" .)