Live comfortably after introduction to duchess (5)
I believe the answer is:
dwell
'live' is the definition.
(to live/dwell somewhere)
'comfortably after introduction to duchess' is the wordplay.
'comfortably' becomes 'well' (synonyms).
'after' means one lot of letters go next to another.
'introduction to' indicates taking the first letters.
The first letter of 'duchess' is 'd'.
'well' put after 'd' is 'DWELL'.
(Other definitions for dwell that I've seen before include "Inhabit" , "Abide, reside" , "Lodge" , "Harp" , "stay" .)