Shrewd when taking a bit of tutelage (6)

I believe the answer is:

astute

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'shrewd' is the definition.
(synonyms)

'when taking a bit of tutelage' is the wordplay.
'when' becomes 'as'.
'a bit of' suggests taking the first letters.
'tutelage' becomes 'ute' (I am not sure about this - if you are sure you should believe this answer much more).
The initial letter of 'taking' is 't'.
'as'+'t'+'ute'='ASTUTE'

Can you help me to learn more?

(Other definitions for astute that I've seen before include "Smart and canny" , "Shrewd and sagacious" , "Subtle" , "Wise" , "Shrewd, wily" .)

I've seen this clue in the Evening Standard.

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