対数関数の導関数

\begin{align*}
y &= \log_{\color{orange}e}\ (\colorbox{mistyrose}{$2x+3$})\\
\\
y' &= \dfrac{1}{\colorbox{mistyrose}{$2x+3$}} \color{orange}\times \dfrac{1}{\log_{e}e} \color{black}\times (\colorbox{mistyrose}{$2x+3$})'\\
\\
&= \dfrac{2}{2x+3} 
\end{align*}

\begin{align*}
y &= x \times \log_{2}x \color{red}\scriptsize\ \ \cdots A \times B 　積の微分！\\
& 　　\ 　　\color{red}\scriptsize A'B + AB'\\
y' &= (x)' \log_{2}x + x(\log_{2}\colorbox{mistyrose}{$x$})'\\
\\
&= 1 \cdot \log_{2}x + x \cdot \dfrac{1}{\colorbox{mistyrose}{$x$}} \times \dfrac{1}{\log_{e}2} \times \colorbox{mistyrose}{$x$}'\\
\\
&= \log_{2}x + \dfrac{1}{\log 2}
\end{align*}

\begin{align*}
y &= \log_{\color{orange}e}\ \colorbox{mistyrose}{$3x$}\\
\\
y' &= \dfrac{1}{\colorbox{mistyrose}{$3x$}} \color{orange}\times \dfrac{1}{\log_{e}e} \color{black}\times (\colorbox{mistyrose}{$3x$})'\\
\\
&= \dfrac{3}{3x} = \dfrac{1}{x} 
\end{align*}

\begin{align*}
y &= \log_{2}\ (\colorbox{mistyrose}{$4x-1$})\\
\\
y' &= \dfrac{1}{\colorbox{mistyrose}{$4x-1$}} \times \dfrac{1}{\log_{e}2} \times (\colorbox{mistyrose}{$4x-1$})'\\
\\
&= \dfrac{4}{(4x-1)\log2} 
\end{align*}

\begin{align*}
y &= \log_{\color{orange}e}\ (\colorbox{mistyrose}{$x^2+1$})\\
\\
y' &= \dfrac{1}{\colorbox{mistyrose}{$x^2+1$}} \color{orange}\times \dfrac{1}{\log_{e}e} \color{black}\times (\colorbox{mistyrose}{$x^2+1$})'\\
\\
&= \dfrac{2x}{x^2+1} 
\end{align*}

\begin{align*}
y &= x \log x -x\\
 &= x \times \log_{}x -x \color{red}\scriptsize\ \ \cdots A \times B 　積の微分！\\
& 　　\ 　　\color{red}\scriptsize A'B + AB'\\
y' &= (x)' \log_{}x + x(\log_{}\colorbox{mistyrose}{$x$})' -(x)'\\
\\
&= 1 \cdot \log_{}x + x \cdot \left(\dfrac{1}{\colorbox{mistyrose}{$x$}} \color{orange}\times \dfrac{1}{\log_{e}e} \times \colorbox{mistyrose}{$x$}'\color{black}\right) -1\\
\\
&= \log_{}x +1-1 = \log_{}x
\end{align*}

\begin{align*}
y &= x \log x -x\\
 &= x \times (\log_{}x -1) \color{red}\scriptsize\ \ \cdots A \times B 　積の微分！\\
& 　　\ 　　\color{red}\scriptsize A'B + AB'\\
y' &= (x)' (\log_{}x-1) + x(\log_{}\colorbox{mistyrose}{$x$}-1)'\\
\\
&= 1 \cdot (\log_{}x-1) + x \cdot \left(\dfrac{1}{\colorbox{mistyrose}{$x$}} \color{orange}\times \dfrac{1}{\log_{e}e} \times \colorbox{mistyrose}{$x$}' -0\right) \\
\\
&= \log_{}x -1+1 = \log_{}x
\end{align*}

\begin{align*}
y &= \log_{\color{orange}e} |\colorbox{mistyrose}{$\cos{x}$}|\\
\\
y' &= \dfrac{1}{\colorbox{mistyrose}{$\cos{x}$}} \color{orange}\times \dfrac{1}{\log_{e}e} \color{black}\times (\colorbox{mistyrose}{$\cos{x}$})'\\
\\
&= \dfrac{-\sin{x}}{\cos{x}} = -\tan{x} 
\end{align*}

\begin{align*}
y &= \log_{3} |\colorbox{mistyrose}{$x^2-1$}|\\
\\
y' &= \dfrac{1}{\colorbox{mistyrose}{$x^2-1$}} \times \dfrac{1}{\log_{e}3} \times (\colorbox{mistyrose}{$x^2-1$})'\\
\\
&= \dfrac{2x}{(x^2-1)\log3} 
\end{align*}

\begin{align*}
y &= \log_{\color{orange}e} |\colorbox{mistyrose}{$3x+2$}|\\
\\
y' &= \dfrac{1}{\colorbox{mistyrose}{$3x+2$}} \color{orange}\times \dfrac{1}{\log_{e}e} \color{black}\times (\colorbox{mistyrose}{$\cos{x}$})'\\
\\
&= \dfrac{-\sin{x}}{\cos{x}} = -\tan{x} 
\end{align*}