Differentiation of a parameter-dependent quadratic form ($d(x^{T}Ax)=(dx{)}^T(A+A^T)x+x^T(dA)x$)

Apply the product rule to the scalar quadratic form and transpose its last scalar term: $x^{T}Adx=(dx{)}^T{A}^{T}x$. Only the symmetric part of $A$ contributes to its value or its derivative with respect to $x$. A varying matrix also contributes the independent term $x^T(dA)x$. No assumption that $A$ is symmetric is needed. Repeating this rule and the chain rule computes mixed derivatives when vector and matrix arguments depend on separate parameters.