## taylor polynomials/derivatives/composite functions

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

and part b) was
find the "k-th" derivative of $sin(x^2)$ at 0 for all k.