求下面函数的偏导数z=(1+xy)^xy

Question

求下面函数的偏导数
z=(1+xy)^xy

Answer 1

d(ln(z))=d(xyln(1+xy))
d(ln(z))=dz/z
d(xyln(1+xy))=xyd(ln(1+xy))+ln(1+xy)d(xy)
=xy(d(xy)/(1+xy))+ln(1+xy)d(xy)
=d(xy)[xy/(1+xy)+ln(1+xy)]
=(xdy+ydx)/[xy/(1+xy)+ln(1+xy)]
由此得z对x的偏导为(1+xy)^xy*y/[xy/(1+xy)+ln(1+xy)]
此得z对y的偏导为(1+xy)^xy*x/[xy/(1+xy)+ln(1+xy)]