A space in which every infinite set contains an infinite subset with only a
finite number of accumulation points is said to have the finite derived
set property. We study this property in the class of spaces in which
compact sets are closed -- the KC-spaces -- and apply our results to
show that among hereditarily Lindel\"of spaces, minimal KC-spaces are
compact. This result generalizes a theorem of \cite{AW} and gives a partial
answer to a question of R. Larson.