设u=u(x),v=v(x)对x都可导
y=uv=u(x)v(x)
按导数的定义,设在x处有改变量t,则y的改变量
Y=u(x+t)v(x+t)-u(x)v(x)
=u(x+t)v(x+t)-u(x)v(x+t) +u(x)v(x+t)-u(x)v(x)
=[u(x+t)-u(x)]*v(t+x) +u(x)*[v(x+t)-v(x)]
Y/t＝v(x+t)*[u(x+t)-u(x)]／t＋u(x)*[v(x+t)-v(x)]／t
当t趋近于零时,v(t+x)的极限是v(x),
u(x+t)-u(x)]／t的极限是u'(x),
[v(x+t)-v(x)]／t的极限是v'(x),
所以有
(uv)' =u'v+uv