Let
$\displaystyle f(x)=\sin x$
$\displaystyle f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$
$\displaystyle =\lim_{h\to 0}\frac{\sin(x+h)-\sin x}{h}$
$\displaystyle =\lim_{h\to 0}\frac{\sin x\cdot\cos h+\cos x\cdot\sin h-\sin x}{h}$
$\displaystyle =\lim_{h\to 0}\left(\sin x\cdot\frac{\cos h-1}{h}+\cos x\cdot\frac{\sin h}{h}\right)$
$\displaystyle =\sin x\cdot\lim_{h\to 0}\frac{\cos h-1}{h}+\cos x\cdot\lim_{h\to 0}\frac{\sin h}{h}$
좌측에
로피탈_정리,L_Hopital_s_rule를 쓰면
$\displaystyle =\sin x\cdot 0+\cos x\cdot 1$
$\displaystyle =\cos x$
i.e.
$\displaystyle \frac{\sin(x+h)-\sin x}{h}$
$\displaystyle =\frac{\sin x\cos h+\cos x\sin h-\sin x}{h}$
$\displaystyle =\sin x\left(\frac{\cos h-1}{h}\right)+\cos x\left(\frac{\sin h}{h}\right)$
$\displaystyle h\to 0$ 일 때의 극한을 구하면
$\displaystyle =\sin x \cdot 0 + \cos x \cdot 1$
$\displaystyle =\cos x$