Suppose that I have a symmetric Toeplitz $n\times n$ matrix $\mathbf{A}=\left[\begin{array}{cccc}a_1&a_2&\cdots& a_n\\a_2&a_1&\cdots&a_{n-1}\\\vdots&\vdots&\ddots&\vdots\\a_n&a_{n-1}&\cdots&a_1\end{array}\right]$ such that $a_i\geq 0$, and a diagonal matrix $\mathbf{B}=\left[\begin{array}{cccc}b_1&0&\cdots& 0\\0&b_2&\cdots&0\\\vdots&\vdots&\ddots&\vdots\\0&0&\cdots&b_n\end{array}\right]$ where $b_i=c/\beta_i$ for some constant $c>0$ such that $\beta_i>0$.
Now let $\mathbf{M}=\mathbf{A}(\mathbf{A}+\mathbf{B})^{-1}\mathbf{A}$.
Can one express a partial derivative $\partial \operatorname{Tr}[\mathbf{M}]\bigg/{\partial\beta_i}$ in closed form, where $\operatorname{Tr}[\mathbf{M}]$ is the trace operator?