Question

对积分上限函数求导时,当上限为函数时需要对该函数求导的原因?

Answer 1

记

F(x) = ∫[0,x]f(t)dt,

有

F‘(x) = f(x),

于是,由于

∫[0,u(x)]f(t)dt = F[u(x)],

可得

d{∫[0,u(x)]f(t)dt}/dx

= dF[u(x)]/dx

= F'[u(x)]*u'(x)

= f[u(x)]*u'(x).