I have a little question about this step in my solution manual
$$ x_j\partial_ka_jx_k - x_j\partial_ka_kx_j = x_ja_j\delta_{kk} -x_ja_k\delta_{kj} $$
the x indices are just from the vector $\textbf{r}=(x_1,x_2,x_3)$ and $\textbf{a}$ is a konstant vector. Anyway, when the del operator have two scalars to the right, can I just choose which one i wanna operate it on? In the expression above we have an $a_j$ and $a_k$ between the del operator and $x$ but they operate on $x$ anyway and pull the $a$ in front? Why can you do that?
Hints:
Your solution manual apparently uses the convention that a differentiation acts on everything to the right of it. Sometimes authors use parenthesis to make that clear.
Note the implicit Einstein notation.
Next use that differentiation is a linear operation.
In particular, one can pull a constant outside of the differentiation.