partial derivative involving a matrix

Question

I wish to know if my calculation is correct. Considering that $A$ is a symmetric matrix.

$$\frac{\partial}{\partial x_{k}}\left(x^{T}Ax\right)=\frac{\partial}{\partial x_{k}}\left(\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}x_{i}x_{j}\right)=\sum_{i=1}^{n}(a_{ik}+a_{ki})x_{i}+2a_{kk}x_{k}=\sum_{i=1}^{n}2a_{ik}x_{i}+2a_{kk}x_{k}=\\2\left(\sum_{i=1}^{n}a_{ik}x_{i}+a_{kk}x_{k}\right)=2\left(Ax\right)_{k}+2a_{kk}x_{k}$$

I am asking since the suggeested evaluation is $2(Ax)_k$ but it might be a mistake.

Thanks!

Answer 1

Hint: Third equality must be like the following, $$ \left( \sum_{i=1,i \neq k}^{n}(a_{ik}+a_{ki})x_{i} \right)+2a_{kk}x_{k} = \sum_{i=1}^{n}(a_{ik}+a_{ki})x_{i}. $$