Background

Help me calculate the triple summation

Problem

We want to show that$$\frac{\partial}{\partial\xi_i}\left[\sum_{i=1}^k\sum_{j=1}^k a_{ij}(\bar{x}_i-\xi_i)(\bar{x}_j-\xi_j)\right] = -2\sum_{j=1}^k a_{ij}(\bar{x}_j-\xi_j),\ i = 1,\ldots, k.$$

What I did

$\forall i$; fixed,$$\frac{\partial}{\partial\xi_i}\left[\sum_{i=1}^k\sum_{j=1}^k a_{ij}(\bar{x}_i-\xi_i)(\bar{x}_j-\xi_j)\right] = -2a_{ii}(\bar{x}_i-\xi_i)-\sum_{j\neq i}^k a_{ij}(\bar{x}_j-\xi_j)$$

What could possibly go wrong? If I am missing information to help you make a decision, please let me know so I can add it.