Chain rule for Hessian. How to compute $D^2 f^\alpha$ How does the chain rule generalize to the Hessian matrix. In particular, how can we compute $$D^2 f^\alpha,$$
where $f:\mathbb{R}^N \to \mathbb{R}$, $N>1$, and $\alpha >0$?

There is nothing tricky about this--you can just use the ordinary single-variable chain rule to compute each partial derivative.  If $\partial_i$ denotes the derivative with respect to the $i$th variable, then $$\partial_i(f^\alpha)=\alpha f^{\alpha-1}\partial_i(f)$$ (this is literally nothing but the fact that for a function $f$ of one variable, the derivative of $f^\alpha$ is $\alpha f^{\alpha-1}f'$).  Then to get a second partial derivative, you just differentiate again the same way (using the product rule and chain rule):
$$\partial_j(\partial_i(f^\alpha))=\partial_j(\alpha f^{\alpha-1}\partial_i(f))=\alpha(\alpha-1)f^{\alpha-2}\partial_j(f)\partial_i(f)+\alpha f^{\alpha-1}\partial_j(\partial_i(f)).$$
