$\displaystyle  \frac{\partial }{\partial \beta } {x}’A^{-1}{x} = $ $\displaystyle  -{x}’A^{-1}\dot{A}_\beta A^{-1}{x}  $    
  $\displaystyle \frac{\partial }{\partial \beta }A^{-1} = $ $\displaystyle  -A^{-1}\dot{A}_\beta A^{-1}  $    
  $\displaystyle \frac{\partial }{\partial \beta } |A| =  $ $\displaystyle  \, \, |A| \, {trace}\left( A^{-1}\dot{A}_\beta \right)  $    
  $\displaystyle \frac{\partial }{\partial \beta } \log \left\{ |A|\right\}  = $ $\displaystyle  \, \, \frac{1}{|A|}\,  \frac{\partial }{\partial \beta }A= {trace}\left( A^{-1}\dot{A}_\beta \right)  $    
  $\displaystyle \frac{\partial ^2}{\partial \beta \partial \theta } A^{-1} = $ $\displaystyle  -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}  $    
  $\displaystyle \frac{\partial ^2}{\partial \beta \partial \theta } \log \left\{ |A|\right\}  = $ $\displaystyle  \, \, {trace}\left(A^{-1}\ddot{A}_{\beta \theta } \right) - {trace}\left(A^{-1}\dot{A}_\beta A^{-1}\dot{A}_\theta \right)  $