= Differentiation of a parameter-dependent quadratic form
{title2=$d(x^TAx)=(dx)^T(A+A^T)x+x^T(dA)x$}
Apply the <product rule> to the scalar <quadratic form> and transpose its last scalar term: $x^TA\,dx=(dx)^TA^Tx$. 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.
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