In general, if $\displaystyle f(x)=g(x^n)$ find the "k-th" derivative of f(x) at 0 in terms of the derivatives of g at 0.

Note, this is part c) of a question in which part a) was

prove the taylor polynomial of $\displaystyle sin(x^2)$ of degree 4n+2 at 0 is

$\displaystyle

(x^2)+...+(-1)^n[x^{4n+2}]/[(2n+1)!]

$ hint: if $\displaystyle P(x)$ is the taylor polynomial of degree 2n+1 for sin(x) at 0 then $\displaystyle sin(x)=P(x)+R(x)$ where the lim as x->0 of $\displaystyle R(x)/(x^{2n+1})$ is 0. what does this imply about the limit as x->0 of $\displaystyle R(x^2)/(x^{4n+2})$ ?

and part b) was

find the "k-th" derivative of $\displaystyle sin(x^2)$ at 0 for all k.