Incorporated one irrational hospital department that's developing (9)
I believe the answer is:
incipient
'developing' is the definition.
(I've seen this in another clue)
'incorporated one irrational hospital department' is the wordplay.
'incorporated' becomes 'inc'.
'one' becomes 'I' (Roman numeral).
'irrational' becomes 'pi' (an irrational number).
'hospital department' becomes 'ENT' (abbreviation for Ear, Nose and Throat department).
'inc'+'i'+'pi'+'ent'='INCIPIENT'
'that's' is the link.
(Other definitions for incipient that I've seen before include "Developing" , "Nascent" , "Beginning to exist, in an early stage" , "Starting to happen" , "Beginning to develop" .)