这个非常基本的问题,如果要是从0开始学的话,要讲好几堂课的.所以
首先具备极限知识的基础：当 x → 0 时,lim sinx /x = 1
其次导数的定义：f'(x) = lim [f(x+Δx) - f(x)]/Δx 当 Δx → 0 时的极限.
(cosx)' = Δx→0 lim [cos(x+Δx) - cosx]/Δx
其中
cos(x+Δx) - cosx
= cos[x + Δx/2 + Δx/2] - cos(x + Δx/2 - Δx/2]
= cos(x+ Δx/2)cos(Δx/2) - sin(x+ Δx/2)sin(Δx/2) - cos(x+ Δx/2)cos(Δx/2) - sin(x + Δx/2)sin(Δx/2)
= -2 sin(x + Δx/2)sin(Δx/2)
(cosx)' = Δx→0 lim -2 sin(x +Δx/2)sin(Δx/2) /Δx
= Δx→0 lim - sin(x + Δx/2) [sin(Δx/2)/(Δx/2)]
= Δx→0 lim - sin(x + Δx/2)
= - sinx