For
$$ f(x) = |x|^\alpha \quad x \in \mathbb{R}^n \setminus \{0\} $$
the first partial derivative is: $$ \begin{align*} \frac{\partial f}{\partial x_i} (x) = \frac{\partial\left( \sum_{k=1}^n x_k^2 \right)^{\frac{\alpha}{2}}}{\partial x_i} &= \frac{\partial}{\partial x_i}(x_1^2 + x_2^k + \cdots + x_i^2 + \cdots + x_n^2)^{\frac{\alpha}{2}} \\ &= \alpha x_i \left(\sum_{k=1}^n x_k^2 \right)^{\frac{\alpha}{2} -1} = \alpha x_i (f(x))^{1 - \frac{2}{\alpha}} \end{align*} $$
How can I get the second order partials i.e.
$$ \frac{\partial f}{\partial x_i \partial x_j} \quad \forall i,j \in \{1, \ldots, n\} $$