微分　 ${\displaystyle {\log }_{a}x}$

${\displaystyle {\left({\log }_{a}x\right)}^{\prime }=\frac{1}{x\log a}}$

■導出

${\displaystyle {\left({\log }_{a}x\right)}^{\prime }}$ ${\displaystyle =\underset{\Delta x\to 0}{\lim }\frac{{\log }_a\left(x+\Delta x\right)-{\log }_a\left(x\right)}{\Delta x}}$

${\displaystyle =\underset{\Delta x\to 0}{\lim }\frac{{\log }_a\left(\frac{x+\Delta x}{x}\right)}{\Delta x}}$

${\displaystyle =\underset{\Delta x\to 0}{\lim }\frac{1}{\Delta x}{\log }_a\left(1+\frac{\Delta x}{x}\right)}$

${\displaystyle \frac{\Delta x}{x}=t}$ とおくと， ${\displaystyle \Delta x=xt}$ ．

また， ${\displaystyle \Delta x\to 0}$ ならば ${\displaystyle t\to 0}$ ．

よって

${\displaystyle =\underset{t\to 0}{\lim }\frac{1}{xt}{\log }_a\left(1+t\right)}$

${\displaystyle =\underset{t\to 0}{\lim }\frac{1}{x}{\log }_a{\left(1+t\right)}^{\frac{1}{t}}}$

${\displaystyle =\frac{1}{x}{\log }_a\left\{\underset{t\to 0}{\lim }{\left(1+t\right)}^{\frac{1}{t}}\right\}}$

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

${\displaystyle =\frac{1}{x}·\frac{\log e}{\log a}}$               (底の変換)

${\displaystyle =\frac{1}{x\log a}}$

■${\displaystyle {\left(\log x\right)}^{\prime }=\frac{1}{x}}$ を利用した方法

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

よって

${\displaystyle {\left(\log x\right)}^{\prime }={\left(\frac{\log x}{\log a}\right)}^{\prime }}$

${\displaystyle =\frac{1}{\log a}{\left(\log x\right)}^{\prime }}$

${\displaystyle =\frac{1}{\log a}·\frac{1}{x}}$

${\displaystyle =\frac{1}{x\log a}}$

 

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最終更新日 2023年6月7日