Proving piecewise function is k-differentiable function

For this problem,

I have a question about the solution,

Do you please know how they go from the $f^{n}(x)$ to $f^{n + 1}(x)$ step (I assume they differentiate $f^{n}(x)$)? I think they made a typo there is something because they did not seem to differentiate the $\frac{1}{x}$ since it is three functions multiplied together so you have to use the product rule on $p_nu$ where $u = \frac{1}{x}e^{-\frac{1}{x}}$.

I also don't understand how they got $p_1(x) = x^2$. Should it not be $p_1(x) = \frac{1}{x}$?

Thanks for any help!