 Solution

Use compatible dimensions $A\in {\mathbb{R}}^{d\times d}$ and $b,x\in {\mathbb{R}}^d$; the printed “b-by-b” size is read consistently with the displayed products and domain. Take derivatives as column gradients. Symmetry of $A$ givesFor example, the directional differential of $x^{T}Ax$ is $d{x}^{T}Ax+x^{T}Adx=2(Ax{)}^{T}dx$. The linear term contributes $b$ to the gradient and zero to the Hessian matrix.