Show, that the function

$$ \mathcal E: \mathbb R \to \mathbb R:
 x \mapsto \begin{cases}
 \exp(-\frac{1}{x^2}), & \text{if x $\neq$ 0}, \\
 0, & \text{otherwise},
 \end{cases} $$

is infinitely differentiable and that $\frac{d^k\mathcal E}{dx}(0) = 0$ for all $k \in \mathbb N$.

We just introduced differentiation, so the solution should not contain very advanced techniques to solve this. We also got the tip it should be done by induction.

@fgp I found that it is
$$f^{(n)}(x) = P\_n \left(\frac1x\right)e^{-\frac1{x^2}}$$
where $P\_n$ is a polynomial with integer coefficients.

I could also do the initial step for the induction:

For $n=1:$
$$f'(x) = \frac2{x^2}e^{-\frac1{x^2}} = P\_1 \left(\frac1t\right) e^{-\frac1{x^2}}$$
where $P\_1(x)=2x^2.$

I am stuck at the induction step: $f^{(n+1)}(x) = $.....