Drama teacher's diligence (10)

I believe the answer is:

stagecoach

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'diligence' is the definition.
(I've seen this before)

'drama teacher's' is the wordplay.
'drama' becomes 'stage' (I can't explain this - if you can you should give a lot more credence to this answer).
'teacher' becomes 'coach' (coach is a kind of teacher).
'stage'+'coach'='STAGECOACH'

Can you help me to learn more?

(Other definitions for stagecoach that I've seen before include "early form of transport" , "Horse-drawn carriage" , "Great Western" , "Diligence" , "Picture of Ford's" .)

I've seen this clue in The Telegraph.

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