$$\begin{eqnarray} \frac{\partial \boldsymbol{ x }^T\boldsymbol{ A }}{\partial \boldsymbol{ x }} &=& \frac{\partial (x_1A_{11}+x_2A_{12}+\cdots+x_nA_{1n}, x_1A_{21}+x_2A_{22}+\cdots+x_nA_{2n},\cdots, x_1A_{m1}+x_2A_{m2}+\cdots+x_nA_{mn}) }{\partial \boldsymbol{ x }} \\ &=& \left( \begin{array}{cccc} \frac{\partial (x_1A_{11}+x_2A_{12}+\cdots)}{\partial x_1} & \frac{\partial (\cdots + x_2A_{12}+x_3A_{13}+\cdots)}{\partial x_2} & \ldots & \frac{\partial (\cdots + x_{n-1}A_{1(n-1)}+x_nA_{1n})}{\partial x_n} \\ \frac{\partial (x_1A_{21}+x_2A_{22}+\cdots)}{\partial x_1} & \frac{\partial (\cdots + x_2A_{22}+x_3A_{23}+\cdots)}{\partial x_2} & \ldots & \frac{\partial (\cdots + x_{n-1}A_{2(n-1)}+x_nA_{2n})}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial (x_1A_{m1}+x_2A_{m2}+\cdots)}{\partial x_1} & \frac{\partial (\cdots + x_2A_{m2}+x_3A_{m3}+\cdots)}{\partial x_2} & \ldots & \frac{\partial (\cdots + x_{n-1}A_{m(n-1)}+x_nA_{mn})}{\partial x_n} \\ \end{array} \right) \\ &=& \left( \begin{array}{cccc} A_{ 11 } & A_{ 12 } & \ldots & A_{ 1n } \\ A_{ 21 } & A_{ 22 } & \ldots & A_{ 2n } \\ \vdots & \vdots & \ddots & \vdots \\ A_{ m1 } & A_{ m2 } & \ldots & A_{ mn } \end{array} \right) \\ &=& \boldsymbol{ A } \end{eqnarray}$$

$$\begin{eqnarray} \frac{\partial \boldsymbol{ A }\boldsymbol{ x }}{\partial \boldsymbol{ x }} &=& \frac{\partial (x_1A_{11}+x_2A_{21}+\cdots+x_nA_{n1}, x_1A_{12}+x_2A_{22}+\cdots+x_nA_{n2},\cdots, x_1A_{1m}+x_2A_{2m}+\cdots+x_nA_{nm}) }{\partial \boldsymbol{ x }} \\ &=& \left( \begin{array}{cccc} \frac{\partial (x_1A_{11}+x_2A_{21}+\cdots)}{\partial x_1} & \frac{\partial (\cdots + x_2A_{21}+x_3A_{31}+\cdots)}{\partial x_2} & \ldots & \frac{\partial (\cdots + x_{n-1}A_{(n-1)1}+x_nA_{n1})}{\partial x_n} \\ \frac{\partial (x_1A_{12}+x_2A_{22}+\cdots)}{\partial x_1} & \frac{\partial (\cdots + x_2A_{22}+x_3A_{32}+\cdots)}{\partial x_2} & \ldots & \frac{\partial (\cdots + x_{n-1}A_{(n-1)2}+x_nA_{n2})}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial (x_1A_{1m}+x_2A_{2m}+\cdots)}{\partial x_1} & \frac{\partial (\cdots + x_2A_{2m}+x_3A_{3m}+\cdots)}{\partial x_2} & \ldots & \frac{\partial (\cdots + x_{n-1}A_{(n-1)m}+x_nA_{nm})}{\partial x_n} \\ \end{array} \right) \\ &=& \left( \begin{array}{cccc} A_{ 11 } & A_{ 21 } & \ldots & A_{ n1 } \\ A_{ 12 } & A_{ 22 } & \ldots & A_{ n2 } \\ \vdots & \vdots & \ddots & \vdots \\ A_{ 1m } & A_{ 2m } & \ldots & A_{ nm } \end{array} \right) \\ &=& \boldsymbol{ A }^T \end{eqnarray}$$

$$\begin{eqnarray} \frac{\partial \boldsymbol{ x }^T \boldsymbol{ A }\boldsymbol{ x }}{\partial \boldsymbol{ x }} &=& \frac{\partial ( \displaystyle\sum_{ i }\displaystyle\sum_{ j } A_{ij}x_ix_j) }{\partial \boldsymbol{ x }} \\ &=& \left( \begin{array}{cccc} \frac{\partial (\displaystyle\sum_{ i }\displaystyle\sum_{ j } A_{ij}x_ix_j)}{\partial x_1} \\ \frac{\partial (\displaystyle\sum_{ i }\displaystyle\sum_{ j } A_{ij}x_ix_j)}{\partial x_2} \\ \vdots \\ \frac{\partial (\displaystyle\sum_{ i }\displaystyle\sum_{ j } A_{ij}x_ix_j)}{\partial x_n} \\ \end{array} \right) \\ &=& \left( \begin{array}{cccc} \displaystyle\sum_{ i }(A_{ 1i }x_i) + \displaystyle\sum_{ i }(A_{ i1 }x_i)\\ \displaystyle\sum_{ i }(A_{ 2i }x_i) + \displaystyle\sum_{ i }(A_{ i2 }x_i)\\ \vdots\\ \displaystyle\sum_{ i }(A_{ ni }x_i) + \displaystyle\sum_{ i }(A_{ in }x_i) \end{array} \right) \\ &=& \boldsymbol{ A }\boldsymbol{x} + \boldsymbol{ A }^T\boldsymbol{x} \\ &=& (\boldsymbol{ A } + \boldsymbol{ A }^T)\boldsymbol{x} \end{eqnarray}$$