$3$\displaystyle{ { $ \log \mid x\mid $ }^{\prime }=\frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}} }$

$3$\displaystyle{\begin{array}{lll}{ $ \log x $ }^{\prime }& \vspace{6}\\ ={ \lim }\limits_{ {\Delta}x\rightarrow 0 }\frac{\hspace{2} \log $ x+{\Delta}x $ - \log x \hspace{2}}{\hspace{2}2\hspace{2}}& \vspace{6}\\ ={ \lim }\limits_{ {\Delta}x\rightarrow 0 }\frac{\hspace{2} \log $ \frac{\hspace{2} x+{\Delta}x \hspace{2}}{\hspace{2}x\hspace{2}} $ \hspace{2}}{\hspace{2} {\Delta}x \hspace{2}}& \vspace{6}\\ ={ \lim }\limits_{ {\Delta}x\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} {\Delta}x \hspace{2}}\log $ 1+\frac{\hspace{2} {\Delta}x \hspace{2}}{\hspace{2}x\hspace{2}} $ & \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}={ \lim }\limits_{ t\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} xt \hspace{2}}\log $ 1+t $ & \vspace{6}\\ ={ \lim }\limits_{ t\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}}\log { $ 1+t $ }^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}} }& \vspace{6}\\ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}}\log \{ { \lim{ $ 1+t $ }^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}} } }\limits_{ t\rightarrow 0 } \} & \vspace{6}\\ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}}\log e& \vspace{6}\\ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}}& \vspace{6}\\ \end{array}}$ $3$\displaystyle{\begin{array}{lll}& \vspace{6}\\ \end{array}}$

公式の導出はPCサイトを参照のこと。

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