Alternative Equation Formats

The equation on this page is available in multiple formats. Depending on your browser, some formats might not render correctly.

MathML

\begin{aligned} \frac{\partial}{\partial β} x^{'} A^{- 1} x = & - x^{'} A^{- 1} {\dot{A}}_{β} A^{- 1} x \\ \frac{\partial}{\partial β} A^{- 1} = & - A^{- 1} {\dot{A}}_{β} A^{- 1} \\ \frac{\partial}{\partial β} | A | = & | A | t r a c e (A^{- 1} {\dot{A}}_{β}) \\ \frac{\partial}{\partial β} \log {| A |} = & \frac{1}{| A |} \frac{\partial}{\partial β} A = t r a c e (A^{- 1} {\dot{A}}_{β}) \\ \frac{\partial^{2}}{\partial β \partial θ} A^{- 1} = & - A^{- 1} {\ddot{A}}_{β θ} A^{- 1} + A^{- 1} {\dot{A}}_{β} A^{- 1} {\dot{A}}_{θ} A^{- 1} + A^{- 1} {\dot{A}}_{θ} A^{- 1} {\dot{A}}_{β} A^{- 1} \\ \frac{\partial^{2}}{\partial β \partial θ} \log {| A |} = & t r a c e (A^{- 1} {\ddot{A}}_{β θ}) - t r a c e (A^{- 1} {\dot{A}}_{β} A^{- 1} {\dot{A}}_{θ}) \end{aligned}

Math Rendered as SVG

Math as LaTeX Source

\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*}