Promote sedative developed without right (9)
I believe the answer is:
advertise
'promote' is the definition.
(I know that promote can be written as advertise)
'sedative developed without right' is the wordplay.
'developed' indicates anagramming the letters.
'without' is an insertion indicator ('without' can be similar in meaning to 'outside').
'right' becomes 'r' (common abbreviation).
'sedative' with letters rearranged gives 'advetise'.
'advetise' enclosing 'r' is 'ADVERTISE'.
(Other definitions for advertise that I've seen before include "Praise - plug" , "Market" , "Publicise, make known" , "Varied set (anag.)" , "Give publicity to" .)