Z=u2v-uv2,u=xcosy,v=xsiny,求αz/αx和αz/αy(微分)Z=u^2• v-u• v^2,u=xcosy,v=xsiny,求αz/αx(注:式中2为指数)求αz/αx和αz/αy

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Z=u2v-uv2,u=xcosy,v=xsiny,求αz/αx和αz/αy(微分)Z=u^2• v-u• v^2,u=xcosy,v=xsiny,求αz/αx(注:式中2为指数)求αz/αx和αz/αy
Z=u2v-uv2,u=xcosy,v=xsiny,求αz/αx和αz/αy(微分)
Z=u^2• v-u• v^2,u=xcosy,v=xsiny,求αz/αx(注:式中2为指数)
求αz/αx和αz/αy

Z=u2v-uv2,u=xcosy,v=xsiny,求αz/αx和αz/αy(微分)Z=u^2• v-u• v^2,u=xcosy,v=xsiny,求αz/αx(注:式中2为指数)求αz/αx和αz/αy
z=x³cos²ysiny-x³cosysin²y
=x³sinycosy(siny+cosy)
∂z/∂x,则把y看成常数
所以∂z/∂x=2x²sinycosy(siny+cosy)
∂z/∂y,把x看成常数
∂z/∂y=x³*[(siny)'cosy(siny+cosy)+siny(cosy)'(siny+cosy)+sinycosy(siny+cosy)']
=x³[cos²y(siny+cosy)-sin²y(siny+cosy)+sinycosy(cosy-siny)]
=x³[(cos²y-sin²y)(siny+cosy)+sinycosy(cosy-siny)]
=x³(cosy-siny)[(siny+cosy)²+sinycosy]
=x³(cosy-siny)(1+3sinycosy)

dz/dx=(2u*u')+u^2*v'-u'* v^2-2uv*v' u'=cosy v'=siny
dz/dy=(2u*u')+u^2*v'-u'* v^2-2uv*v' u'=-xsiny v'=xcosy
希望你看得懂,我就简单写一下,实际写的时候u' v'要写成对应x、y的导数形式

如下

dz/dx=(2u*u')+u^2*v'-u'* v^2-2uv*v' u'=cosy v'=siny
dz/dy=(2u*u')+u^2*v'-u'* v^2-2uv*v' u'=-xsiny v'=xcosy
z=x³cos²ysiny-x³cosysin²y
=x³sinycosy(siny+cosy)
...

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dz/dx=(2u*u')+u^2*v'-u'* v^2-2uv*v' u'=cosy v'=siny
dz/dy=(2u*u')+u^2*v'-u'* v^2-2uv*v' u'=-xsiny v'=xcosy
z=x³cos²ysiny-x³cosysin²y
=x³sinycosy(siny+cosy)
∂z/∂x,则把y看成常数
所以∂z/∂x=2x²sinycosy(siny+cosy)
∂z/∂y,把x看成常数
∂z/∂y=x³*[(siny)'cosy(siny+cosy)+siny(cosy)'(siny+cosy)+sinycosy(siny+cosy)']
=x³[cos²y(siny+cosy)-sin²y(siny+cosy)+sinycosy(cosy-siny)]
=x³[(cos²y-sin²y)(siny+cosy)+sinycosy(cosy-siny)]
=x³(cosy-siny)[(siny+cosy)²+sinycosy]
=x³(cosy-siny)(1+3sinycosy)

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