Derivative of $(A+BC^{-T}B^T)^{-1}BC^{-1}$

Question

Suppose $A:\mathbb{R}\rightarrow\mathbb{R}^{m\times m}$, $B:\mathbb{R}\rightarrow\mathbb{R}^{m\times n}$, and $C:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$. I'm trying to find the derivative with respect to a scalar $x$ of \begin{equation}(A+BC^{-T}B^T)^{-1}BC^{-1}\end{equation} My matrix calculus is very rusty. I guess we can set \begin{equation}\phi=A+BC^{-T}B^T\end{equation} so that we have \begin{align} \frac{\partial(\phi^{-1} B C^{-1})}{\partial x}&=\frac{\partial\phi^{-1}}{\partial x}B C^{-1} + \phi^{-1}\frac{\partial B}{\partial x}C^{-1} + \phi^{-1}B\frac{\partial C^{-1}}{\partial x}\\ &=\frac{\partial\phi^{-1}}{\partial x}B C^{-1} + \phi^{-1}\frac{\partial B}{\partial x}C^{-1} - \phi^{-1}BC^{-1}\frac{\partial C}{\partial x}C^{-1}. \end{align} Now, we only need to find $\frac{\partial\phi^{-1}}{\partial x}$. That is, \begin{equation}\frac{\partial}{\partial x}\left(A+BC^{-T}B^T\right)^{-1}\end{equation} Is the following correct? \begin{equation}\frac{\partial}{\partial x}\left(A+BC^{-T}B^T\right)^{-1}=-\left(A+BC^{-T}B^T\right)^{-1}\left[\frac{\partial}{\partial x}\left(A+BC^{-T}B^T\right)\right] \left(A+BC^{-T}B^T\right)^{-1}\end{equation} where \begin{align}\frac{\partial}{\partial x}\left(A+BC^{-T}B^T\right)&=\frac{\partial A}{\partial x}+\frac{\partial B}{\partial x}C^{-T}B^T+B\frac{\partial C^{-T}}{\partial x}B^T+BC^{-T}\frac{\partial B^T}{\partial x}\\ &=\frac{\partial A}{\partial x}+\frac{\partial B}{\partial x}C^{-T}B^T-BC^{-T}\left(\frac{\partial C}{\partial x}\right)^{T}C^{-T}B^T+BC^{-T}\left(\frac{\partial B}{\partial x}\right)^T\end{align}