I need some help figuring out the following proof that my professor put in my notes...

Let $\displaystyle f(x)=e^{-1/x^2}, x\not=0$. Then, $\displaystyle f'(x)=\frac{2}{x^3}f(x), f''(x)=(\frac{-6}{x^4}+\frac{2}{x^3})f(x)$. Prove by induction that $\displaystyle f^n(x)=q_n\frac{1}{x}f(x)$ where $\displaystyle q_n$ is a polynomial.

This is pretty easy to see why it works as we can continue the product rule forever here and continue to get f(x) times a polynomial. However, I'm not sure how to prove it by induction. Any help would be appreciated.