Question-148783

Question Number 148783 by 0731619 last updated on 31/Jul/21

Commented by hknkrc46 last updated on 31/Jul/21

(1) \lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

= \lim_{h \to 0} \frac{- \cos (x + h) + \cos x}{h}

= \lim_{h \to 0} \frac{\cos x - \cos x \cdot \cos h + \sin x \cdot \sin h}{h}

= \lim_{h \to 0} \frac{\cos x (1 - \cos h)}{h} + \lim_{h \to 0} \frac{\sin x \cdot \sin h}{h}

= \cos x \cdot \lim_{h \to 0} \frac{1 - \cos h}{h} + \sin x \cdot \lim_{h \to 0} \frac{\sin h}{h}

= \cos x \cdot 0 + \sin x \cdot 1 = \sin x

★ \frac{d (- \cos x)}{d x} = \sin x \Leftrightarrow \int \sin x d x = - \cos x + c