Taylor's Theorem problem 1

Let $\displaystyle g(x) = e^{\frac{-1}{x^{2}}}$ for $\displaystyle x \ne 0$ and $\displaystyle g(0) = 0$.

Show that $\displaystyle g^{n}(0) = 0$ for all $\displaystyle n\ge 0$ and show that the Taylor series for g about 0 agrees with $\displaystyle g$ only at $\displaystyle x=0$.

So yeah, I am completely lost now in my real analysis class and really don't understand much of anything going on now. It says to use induction to prove that there exist polynomials $\displaystyle p_{kn}$, $\displaystyle 1\le k \le n$ so that $\displaystyle g^{(n)}(x) = \sum_{k=1}^{n}f^{(k)}(x^{2})p_{kn}(x)$ for $\displaystyle x\in \mathbb{R}, n\ge 1$, where $\displaystyle f(x)=e^{\frac{-1}{x}}$. Don't really now where to begin, thanks.