题目

一道二阶偏导数
Z=e^xy乘sin xy 的四个二阶偏导数.

提问时间：2021-02-23

答案

z=e^(xy)*sin(xy)
dz/dx=ye^(xy)*sin(xy)+ye^(xy)*cos(xy)=ye^(xy)*[sin(xy)+cos(xy)]
dz/dy=xe^(xy)*sin(xy)+xe^(xy)*cos(xy)=xe^(xy)*[sin(xy)+cos(xy)]
d²z/dx²=y²e^(xy)[sin(xy)+cos(xy)]+ye^(xy)[ycos(xy)-ysin(xy)]
=2y²e^(xy)*cos(xy)
d²z/dy²=x²e^(xy)[sin(xy)+cos(xy)]+xe^(xy)[xcos(xy)-xsin(xy)]
=2x²e^(xy)*cos(xy)
d²z/dxdy=[ye^(xy)]'*[sin(xy)+cos(xy)]+ye^(xy)*[sin(xy)+cos(xy)]'
=[(1+x)e^(xy)][sin(xy)+cos(xy)]+ye^(xy)*[xcos(xy)-xsin(xy)]
=e^(xy)*{(1+x)[sin(xy)+cos(xy)+xy[cos(xy)-sin(xy)]}
=d²z/dydx
（偏导符打不出来,用d代替）