$$\displaystyle\frac{\partial} {\partial \boldsymbol{x}} \boldsymbol{U}^T \boldsymbol{x}=\boldsymbol{U} $$
$$\displaystyle\frac{\partial} {\partial \boldsymbol{x}} \boldsymbol{x}^T \boldsymbol{U}=\boldsymbol{U} $$
$$\displaystyle\frac{\partial} {\partial \boldsymbol{x}} \boldsymbol{x}^T \boldsymbol{U} \boldsymbol{x}=2\boldsymbol{Ux}$$
$$\displaystyle\frac{\partial} {\partial \boldsymbol{x}} \boldsymbol{U} \boldsymbol{x}=\boldsymbol{U}^T $$