Detective in study (3)
I believe the answer is:
eye
I believe this is a double definition.
'detective' is the first definition.
(as in a private eye)
'study' is the second definition.
(both can mean to examine)
'in' acts as a link.
(Other definitions for eye that I've seen before include "'Cast a cold ... on life, on death' (Yeats)" , "Look at; centre" , "Observe narrowly" , "Dark spot on a potato" , "Body organ" .)