Proof of $\frac{\partial}{\partial x_l}(a_{ijk}x_{i}x_{j}x_{k})$ where $a_{ijk}$ are constant

Solution 1:

Your answer and the actual answer are both correct and equivalent. The only issue is that the "actual" answer relabeled the dummy indices to have a more compact expression.

To elaborate—note that, in each term of your answer, you have a different set of dummy indices you're summing over. The $a_{ijl}x_{i}x_{j}$ term has $i$ and $j$, the $a_{ilk}x_{i}x_{k}$ term has $i$ and $k$, and $a_{ljk}x_{j}x_{k}$ has $j$ and $k$. Because these are dummy indices, and each term is independent of the other, we can replace those dummy indices with whatever we want (as long as they have the same "dimension").

Assuming this, you arrive at the "actual" answer by replacing the $k$ with $j$ in the $a_{ilk}x_{i}x_{k}$ term, and replacing $j$ with $i$ and $k$ with $j$ in the $a_{ljk}x_{j}x_{k}$ term.