Question

若f(x)的一个原函数为sinx，求不定积分f(x)的一阶导数（不定积分求出答案是cosx）过程是？

Answer 1

∫ ƒ(x) dx = sinx
d/dx ∫ ƒ(x) dx = d/dx sinx dx
ƒ(x) = cosx
直接用了公式，∫ cosx dx = sinx + C，即sinx的导数是cosx

过程要由导数推导：dy/dx = lim(Δx→0) [f(x + Δx) - f(x)]/Δx
y = sinx
dy/dx = lim(Δx→0) [sin(x + Δx) - sinx]/Δx
= lim(Δx→0) [(1/2)cos((x + Δx + x)/2)sin((x + Δx - x)/2)]/Δx
= lim(Δx→0) [(1/2)cos(x + Δx/2)sin(Δx/2)]/Δx
= lim(Δx→0) cos(x + Δx/2) · lim(Δx→0) sin(Δx/2)/(Δx/2)
= cos(x + 0) · ( 1 )
= cosx
∴(sinx)' = cosx ==> ∫ cosx dx = sinx + C