I'm not sure what you mean by leading coefficient; leading coefficient of what?
Since the set is a basis for , the first equation you have implies each is 0.
given that { f(x), f' (x)..., f^n (x)} is a basis for Pn,
then a_0 f(x)+ a_1 f'(x)+...a_n f^n (x)=0 , hw can you tell that the leading coefficient is a_0 a_n and that
for a_1 f'(x)+...a_n f^n (x)=0 where a_0 f(x)=0 then the leading coefficient is 2a_1 a_n?