2. Derivative of Sine
Proof that
𝒅
𝒅𝒙
𝐬𝐢𝐧 𝒙 = 𝐜𝐨𝐬 𝒙
𝒅
𝒅𝒙
sin 𝒙 = lim
𝒉→𝟎
𝒇 𝒙+𝒉 −𝒇(𝒙)
𝒉
(limit definition)
𝒅
𝒅𝒙
sin 𝒙 = lim
𝒉→𝟎
sin 𝒙+𝒉 −sin 𝒙
𝒉
(plugging in f(x) = sin 𝑥)
𝒅
𝒅𝒙
sin 𝒙 = lim
𝒉→𝟎
sin 𝒙 cos 𝒉+cos 𝒙 sin 𝒉−sin 𝒙
𝒉
Rule: sin 𝐴 + 𝐵 = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵
𝒅
𝒅𝒙
sin 𝒙 = lim
𝒉→𝟎
− sin 𝒙(𝟏+cos 𝒉)+cos 𝒙 sin 𝒉
𝒉
(factored out-sin 𝑥)
𝒅
𝒅𝒙
sin 𝒙 = − sin 𝒙(lim
𝒉→𝟎
𝟏−cos 𝒉
𝒉
) + cos 𝒙(lim
𝒉→𝟎
sin 𝒉
𝒉
) Rule: lim
𝑥→0
1−cos 𝑥
𝑥
= 𝟎; lim
𝒙→𝟎
sin 𝒙
𝒙
= 𝟏
𝒅
𝒅𝒙
sin 𝒙 = − sin 𝒙 𝟎 + cos 𝒙 (𝟏)
𝒅
𝒅𝒙
sin 𝒙 = cos 𝒙