\documentclass{article}
\usepackage{amsmath}
\newcommand\mb{}
\newcommand\mr{}
\newcommand\bA{A}
\begin{document}

\begin{align*}
   \frac{\partial}{\partial\beta} \mb{x}'\bA^{-1}\mb{x} =&
      -\mb{x}'\bA^{-1}\dot{\bA}_\beta\bA^{-1}\mb{x} \\
   \frac{\partial}{\partial\beta}\bA^{-1} =&
      -\bA^{-1}\dot{\bA}_\beta\bA^{-1}  \\
   \frac{\partial}{\partial \beta} |\bA| = &
       \,\,|\bA| \,\mr{trace}\left( \bA^{-1}\dot{\bA}_\beta \right) \\
   \frac{\partial}{\partial \beta} \log\left\{|\bA|\right\} =&
       \,\,\frac{1}{|\bA|}\, \frac{\partial}{\partial\beta}\bA =
       \mr{trace}\left( \bA^{-1}\dot{\bA}_\beta \right) \\
   \frac{\partial^2}{\partial\beta\partial\theta} \bA^{-1} =&
       -\bA^{-1}\ddot{\bA}_{\beta\theta}\bA^{-1} +
        \bA^{-1}\dot{\bA}_\beta \bA^{-1}\dot{\bA}_\theta\bA^{-1} +
        \bA^{-1}\dot{\bA}_\theta\bA^{-1}\dot{\bA}_\beta \bA^{-1} \\
   \frac{\partial^2}{\partial\beta\partial\theta} \log\left\{|\bA|\right\} =&
        \,\,\mr{trace}\left(\bA^{-1}\ddot{\bA}_{\beta\theta} \right)  -
        \mr{trace}\left(\bA^{-1}\dot{\bA}_\beta\bA^{-1}\dot{\bA}_\theta\right)
\end{align*}

\end{document}
