For each child, a line that's derogatory (8)
I believe the answer is:
personal
'derogatory' is the definition.
(thesaurus)
'for each child a line' is the wordplay.
'for each' becomes 'per' (I've seen this before).
'child' becomes 'son' (son is a kind of child**).
'line' becomes 'l'.
'per'+'son'+'a'+'l'='PERSONAL'
'that's' acts as a link.
(Other definitions for personal that I've seen before include "Individual; advert (column)" , "Concerning the self" , "Concerning oneself" , "Distinctive; individual" , "Applied to a particular individual" .)