Z=u2v-uv2,u=xcosy,v=xsiny,求αz/αx和αz/αy(微分)
问题描述：
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
答
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)