자연로그 미분 증명 ¶
$\displaystyle \frac{d}{dx}\ln x$
$\displaystyle =\lim_{h\to 0}\frac{\ln(x+h)-\ln x}{h}$
$\displaystyle =\lim_{h\to 0}\frac{\ln\left(\frac{x+h}{x}\right)}{h}$
$\displaystyle =\lim_{h\to 0}\frac{\ln\left(1+\frac{h}{x}\right)}{h}$
$\displaystyle =\frac1x\lim_{h\to 0}\frac{\ln\left( 1+\frac{h}{x} \right)}{\frac{h}{x}}$
$\displaystyle =\frac1x\lim_{h\to 0}\ln\left( 1+\frac{h}{x} \right)^{\frac{x}{h}}$
$\displaystyle =\frac1x\ln e$
$\displaystyle =\frac1x$
Up: 여러가지증명$\displaystyle =\lim_{h\to 0}\frac{\ln\left(\frac{x+h}{x}\right)}{h}$
$\displaystyle =\lim_{h\to 0}\frac{\ln\left(1+\frac{h}{x}\right)}{h}$
$\displaystyle =\frac1x\lim_{h\to 0}\frac{\ln\left( 1+\frac{h}{x} \right)}{\frac{h}{x}}$
$\displaystyle =\frac1x\lim_{h\to 0}\ln\left( 1+\frac{h}{x} \right)^{\frac{x}{h}}$
$\displaystyle =\frac1x\ln e$
$\displaystyle =\frac1x$