2×2行列で試してみます．
$$\mbox{def}\,\,\,\,\,\boldsymbol{w}
=\left[\begin{array}{cc}
w_{11} & w_{12} \\
w_{21} & w_{22}
\end{array} \right]
\,\,\,\,\,\,\, \mbox{subject to} \,\,\mbox{det} \boldsymbol{w}>0$$
■$\displaystyle\frac{\partial} {\partial \boldsymbol{w}}\mbox{log}(\mbox{det}\boldsymbol{w})$
$$\displaystyle\frac{\partial} {\partial \boldsymbol{w}}\mbox{log}(\mbox{det}\boldsymbol{w})
=\left[\begin{array}{cc}
\displaystyle\frac{\partial} {\partial w_{11}}\mbox{log}(\mbox{det}\boldsymbol{w}) & \displaystyle\frac{\partial} {\partial w_{12}}\mbox{log}(\mbox{det}\boldsymbol{w}) \\
\ \displaystyle\frac{\partial} {\partial w_{21}}\mbox{log}(\mbox{det}\boldsymbol{w}) & \displaystyle\frac{\partial} {\partial w_{22}}\mbox{log}(\mbox{det}\boldsymbol{w})
\end{array}
\right]$$
$$=\displaystyle\frac{1}{\mbox{det}\boldsymbol{w}} \left[
\begin{array}{cc} w_{22} & -w_{21} \\
-w_{12} & w_{11}
\end{array} \right]= \underline{(\boldsymbol{w}^{T})^{-1}}$$
■$\displaystyle\frac{\partial} {\partial x} \mbox{log}(\mbox{det}\boldsymbol{w})$
$$\displaystyle\frac{\partial} {\partial x} \mbox{log}(\mbox{det}\boldsymbol{w})
=\displaystyle\frac{1}{\mbox{det}\boldsymbol{w}}(\displaystyle\frac{\partial}{\partial w_{11}}\mbox{det}\boldsymbol{w}\cdot \displaystyle\frac{\partial w_{11}} {\partial x}+\displaystyle\frac{\partial}{\partial w_{12}}\mbox{det}\boldsymbol{w}\cdot \displaystyle\frac{\partial w_{12}} {\partial x}+\displaystyle\frac{\partial}{\partial w_{21}}\mbox{det}\boldsymbol{w}\cdot \displaystyle\frac{\partial w_{21}} {\partial x}+\displaystyle\frac{\partial}{\partial w_{22}}\mbox{det}\boldsymbol{w}\cdot \displaystyle\frac{\partial w_{22}} {\partial x})$$
$$=\displaystyle\frac{1}{\mbox{det}\boldsymbol{w}}(w_{22} \displaystyle\frac{\partial w_{11}} {\partial x}-w_{21}\displaystyle\frac{\partial w_{12}} {\partial x}-w_{12}\displaystyle\frac{\partial w_{21}} {\partial x}+w_{11}\displaystyle\frac{\partial w_{22}} {\partial x})$$
$$=\mbox{trace} \left[ \displaystyle\frac{1}{\mbox{det}\boldsymbol{w}}\left
[\begin{array}{cc}
w_{22} & -w_{12} \\
-w_{21} & w_{11}
\end{array} \right]
\left[\begin{array}{cc}
\displaystyle\frac{\partial w_{11}} {\partial x} & \displaystyle\frac{\partial w_{12}} {\partial x} \\
\displaystyle\frac{\partial w_{21}} {\partial x} & \displaystyle\frac{\partial w_{22}} {\partial x}
\end{array} \right] \right]
=\underline{\mbox{trace} \left(\boldsymbol{w}^{-1}\displaystyle\frac{\partial \boldsymbol{w}}{\partial x}\right)}$$