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

Commented byhknkrc46 last updated on 31/Jul/21

(1) lim_(h → 0) ((f(x + h) − f(x))/h) = lim_(h → 0) ((−cos (x + h) + cos x)/h) = lim_(h → 0) ((cos x − cos x ∙ cos h + sin x ∙ sin h)/h) = lim_(h → 0) ((cos x(1 − cos h))/h) + lim_(h → 0) ((sin x ∙ sin h)/h) = cos x ∙lim_(h → 0) ((1 − cos h)/h) + sin x ∙lim_(h → 0) ((sin h)/h) = cos x ∙ 0 + sin x ∙ 1 = sin x ★ ((d(−cos x))/dx) = sin x ⇔ ∫sin x dx = −cos x + c