ガウス分布の\(\bSigma\)についての微分
\[
\small
\begin{align}
\pdiff{\N}{\bSigma}
&=\pdiff{}{\bSigma}{1\over(2\pi)^{D/2}}{1\over|\bSigma|^{1/2}}\Gexp \\
&={1\over(2\pi)^{D/2}} \l(\pdiff{}{\bSigma}{1\over|\bSigma|^{1/2}}\r)\Gexp \\
&~~~~+{1\over(2\pi)^{D/2}}{1\over|\bSigma|^{1/2}}\pdiff{}{\bSigma}\Gexp~~~\cmt{※3} \\
&={1\over(2\pi)^{D/2}}\l(-{1\over2}\r)|\bSigma|^{-1/2}\bSigmai\Gexp~~~\cmt{※4} \\
&~~~~+{1\over(2\pi)^{D/2}}{1\over|\bSigma|^{1/2}}{1\over2}\Gexp
\bSigmai(\bx-\bmu)(\bx-\bmu)^\T\bSigmai~~~\cmt{※5} \\
&=-{1\over2}\bSigmai\N(\bx|\bmu,\bSigma)
+{1\over2}\bSigmai(\bx-\bmu)(\bx-\bmu)^\T\bSigmai\N(\bx|\bmu,\bSigma) \\
&=-{1\over2}\l\{\bSigmai-\bSigmai(\bx-\bmu)(\bx-\bmu)^\T\bSigmai\r\}
\N(\bx|\bmu,\bSigma) \\
\end{align}
\]