指数関数の導関数

\large
(a^{\color{red}{\large ■}})' = a^{\color{red}{\large ■}} \times \log_{e}a \times (\textcolor{red}{\large ■})'

\begin{align*}
(e^{\color{red}{\large ■}})' = e^{\color{red}{\large ■}} \times \log_{e}e \times (\textcolor{red}{\large ■})'
\end{align*}

\large
(e^{\color{red}{\large ■}})' = e^{\color{red}{\large ■}} \times (\textcolor{red}{\large ■})'

\begin{align*}
y &= e^{\colorbox{mistyrose}{$\footnotesize 3x$}}\\
& 　\color{red}\scriptsize \Darr そのまま！\\
y' &= e^{3x} \textcolor{orange}{\times \log_{e}e} \times(\colorbox{mistyrose}{$3x$})'\\
&= e^{3x} \textcolor{orange}{\times 1}\times 3\\
&=3 e^{3x}

\end{align*}

\begin{align*}
y &= x \times 2^{x}　\color{red}\scriptsize A \times B　積の微分！\\
& 　\color{red}\scriptsize 　　　\ \ A'B+AB'\\
y' &= (x)' \cdot2^{x}+x \cdot (2^x)'\\
&= 1 \cdot 2^x+x \cdot2^{x} \times \log_{e}2 \color{orange}\times x'\\
&= 2^x+x \cdot2^{x} \cdot \log_{e}2\\
&=2^x \left( 1+x\log_{e}2 \right)
\end{align*}

\begin{align*}
y &= e^{\colorbox{mistyrose}{$\footnotesize 2x$}}\\
& 　\color{red}\scriptsize \Darr そのまま！\\
y' &= e^{2x} \textcolor{orange}{\times \log_{e}e} \times(\colorbox{mistyrose}{$2x$})'\\
&= e^{2x} \textcolor{orange}{\times 1}\times 2\\
&=2 e^{2x}

\end{align*}

\begin{align*}
y &= e^{\colorbox{mistyrose}{$\footnotesize -x^2$}}\\
& 　\color{red}\scriptsize \Darr そのまま！\\
y' &= e^{-x^2} \textcolor{orange}{\times \log_{e}e} \times(\colorbox{mistyrose}{$-x^2$})'\\
&= e^{-x^2} \textcolor{orange}{\times 1}\times (-2x)\\
&=-2x e^{-x^2}

\end{align*}

\begin{align*}
y &= 3^{\colorbox{mistyrose}{$\footnotesize x$}}\\
& 　\color{red}\scriptsize \Darr そのまま！\\
y' &= 3^{x} \times \log_{e}3 \color{orange}\times(\colorbox{mistyrose}{$x$})'\\
&= 3^{x} \times \log_{e}3 \textcolor{orange}{\times 1}\\
&=3x \log_{}3

\end{align*}

\begin{align*}
y &= 2^{\colorbox{mistyrose}{$\footnotesize -3x$}}\\
& 　\color{red}\scriptsize \Darr そのまま！\\
y' &= 2^{-3x} \times \log_{e}2 \color{orange}\times(\colorbox{mistyrose}{$-3x$})'\\
&= 2^{-3x} \times \log_{e}2 \times (-3)\\
&=-3 \cdot 2^{-3x} \log_{}2

\end{align*}

\begin{align*}
y &= x \times e^{x}　\color{red}\scriptsize A \times B　積の微分！\\
& 　\color{red}\scriptsize 　　　\ \ A'B+AB'\\
y' &= (x)' \cdot e^{x}+x \cdot (e^x)'\\
&= 1 \cdot e^x+x \cdot e^{x} \color{orange} \times \log_{e}e\times x'\\
&= e^x+x \cdot e^{x} \color{orange}\times 1 \times 1\\
&=e^x \left( 1+x \right)
\end{align*}

\begin{align*}
y &= (2x-1) \times a^{x}　\color{red}\scriptsize A \times B　積の微分！\\
& 　\color{red}\scriptsize 　　　\ \ A'B+AB'\\
y' &= (2x-1)' \cdot a^{x}+(2x-1) \cdot (a^x)'\\
&= 2 \cdot a^x+(2x-1) \cdot a^{x} \times \log_{e}a \color{orange} \times x'\\
&= 2 a^x+(2x-1) \cdot a^{x} \log_{e}a \color{orange}\times 1\\
&=a^x \left\{ 2+(2x-1)\log_{}a \right\}
\end{align*}