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# 微分$3\displaystyle{\hspace{1}{\log }_{a}x}$

$3\displaystyle{{ \( \hspace{1}{ \log }_{a}x \) }^{\prime }=\frac{\hspace{2}1\hspace{2}}{\hspace{2} x\log a \hspace{2}}}$

■導出

$3\displaystyle{{ \( \hspace{1}{ \log }_{a}x \) }^{\prime }}$ $3\displaystyle{={ \lim }\limits_{ {\Delta}x\rightarrow 0 }\frac{\hspace{2} \hspace{1}{ \log }_{a} \( x+{\Delta}x \) - \hspace{1}{ \log }_{a} \(x\) \hspace{2}}{\hspace{2} {\Delta}x \hspace{2}}}$

$3\displaystyle{={ \lim }\limits_{ {\Delta}x\rightarrow 0 }\frac{\hspace{2} \hspace{1}{ \log }_{a} \( \frac{\hspace{2} x+{\Delta}x \hspace{2}}{\hspace{2}x\hspace{2}} \) \hspace{2}}{\hspace{2} {\Delta}x \hspace{2}}}$

$3\displaystyle{ ={ \lim }\limits_{ {\Delta}x\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} {\Delta}x \hspace{2}}\hspace{1}{ \log }_{a} \( 1+\frac{\hspace{2} {\Delta}x \hspace{2}}{\hspace{2}x\hspace{2}} \) }$

$3\displaystyle{\frac{\hspace{2} {\Delta}x \hspace{2}}{\hspace{2}x\hspace{2}}=t}$ とおくと， $3\displaystyle{{\Delta}x=xt}$

また， $3\displaystyle{{\Delta}x\rightarrow 0}$ ならば $3\displaystyle{t\rightarrow 0}$

よって，

$3\displaystyle{={ \lim }\limits_{ t\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} xt \hspace{2}}\hspace{1}{ \log }_{a} \( 1+t \) }$

$3\displaystyle{={ \lim }\limits_{ t\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}}\hspace{1}{ \log }_{a}{ \( 1+t \) }^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}} }}$

$3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}}\hspace{1}{ \log }_{a} \{ { \lim }\limits_{ t\rightarrow 0 }{ \( 1+t \) }^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}} } \} }$

$3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}}\hspace{1}{ \log }_{a}e}$               (∵"e "の定義)

$3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}}\cdot \frac{\hspace{2} \log e \hspace{2}}{\hspace{2} \log a \hspace{2}}}$               (底の変換)

$3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} x\log a \hspace{2}}}$

$3\displaystyle{{ \( \log x \) }^{\prime }=\frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}}}$ を利用した方法

$3\displaystyle{\hspace{1}{\log }_{a}x=\frac{\hspace{2} \log x \hspace{2}}{\hspace{2} \log a \hspace{2}}}$   (底を$3\displaystyle{e}$に変換)

よって

$3\displaystyle{{ \( \log x \) }^{\prime }={ \( \frac{\hspace{2} \log x \hspace{2}}{\hspace{2} \log a \hspace{2}} \) }^{\prime }}$

$3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} \log a \hspace{2}}{ \( \log x \) }^{\prime }}$

$3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} \log a \hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}}}$

$3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} x\log a \hspace{2}}}$

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