Divine female's initially extremely bad repute (7)
I believe the answer is:
euterpe
'divine female's' is the definition.
(one of the nine muses in Greek mythology)
'initially extremely bad repute' is the wordplay.
'initially' suggests taking the first letters.
'bad' indicates an anagram.
The initial letter of 'extremely' is 'e'.
'repute' is an anagram of 'uterpe'.
'e'+'uterpe'='EUTERPE'
(Other definitions for euterpe that I've seen before include "inspiring female" , "Muse of lyric poetry and music" , "Inspiring goddess" , "One of the Muses" , "Musical Muse" .)