Matrix calculus: derivative of the form $x^T A y + y^T A x$

Question

Consider the column vectors $\boldsymbol{x}(z),\boldsymbol{y}(z) \in \mathbb{R^n}$ and a matrix $\boldsymbol{A}(z) \in \mathbb{R}^{n \times n}$, which are all differentiable functions of $z \in \mathbb{R}$.

Determine $\frac{\partial}{\partial z} (\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{y} + \boldsymbol{y}^T\boldsymbol{A}\boldsymbol{x})$. Is there any known form which can be related to it?

Answer 1

We can use product rule,

\begin{align} \frac{\partial }{\partial z} (x^TAy + y^TAx) &=\frac{\partial }{\partial z} (x^T(A+A^T)y )\\ &=\left(\frac{\partial x}{\partial z} \right)^T (A+A^T)y + x^T\left(\frac{\partial A}{\partial z} + \frac{\partial A^T}{\partial z}\right)y + x^T(A+A^T)\frac{\partial y}{\partial z} \end{align}