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