# 微分${\mathrm{log}}_{a}x$

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

■導出

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

$=\underset{\Delta x\to 0}{\mathrm{lim}}\frac{{\mathrm{log}}_{a}\left(\frac{x+\Delta x}{x}\right)}{\Delta x}$

$=\underset{\Delta x\to 0}{\mathrm{lim}}\frac{1}{\Delta x}{\mathrm{log}}_{a}\left(1+\frac{\Delta x}{x}\right)$

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

また， $\Delta x\to 0$ ならば $t\to 0$

よって

$=\underset{t\to 0}{\mathrm{lim}}\frac{1}{xt}{\mathrm{log}}_{a}\left(1+t\right)$

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

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

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

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

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

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

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

よって

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

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

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

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

ホーム>>カテゴリー分類>>指数/対数>>基本となる関数の導関数>>微分　log[a]x