One beginning to smile after actual education's completed (8)
I believe the answer is:
realised
'completed' is the definition.
(I know that completed can be written as realised)
'one beginning to smile after actual education's' is the wordplay.
'one' becomes 'i' (Roman numeral).
'beginning to' suggests taking the first letters.
'after' means one lot of letters go next to another.
'actual' becomes 'real' (similar in meaning).
'education' becomes 'ed' (abbreviation - e.g. sex ed).
The initial letter of 'smile' is 's'.
'i'+'s'='is'
'is' after 'real' is 'realis'.
'realis'+'ed'='REALISED'
(Other definitions for realised that I've seen before include "Came to understand" , "Became fully aware of or converted into money" , "Sidereal (anag)" , "Came to awareness" , "Fully comprehended" .)