u（x)=lnx,v(x)=e^x求(uv)的三阶微分

Question

u（x)=lnx,v(x)=e^x 求(uv)的三阶微分

Answer 1

u=lnx
u'= 1/x
u'' = -1/x^2
u'''= 2/x^3
v = e^x
v'=v''=v'''=e^x
(uv)' =uv'+u'v
(uv)'' = uv''+2u'v'+u''v
(uv)''' = uv'''+3u'v''+3u''v'+u'''v
=(lnx)e^x+ 3e^x/x- 3e^x/x^2 + 2e^x/x^3