设Z的X次方=Y的Z次方所确定的隐函数Z=Z(X,Y),求DZ
问题描述:
设Z的X次方=Y的Z次方所确定的隐函数Z=Z(X,Y),求DZ
答
z^x=y^z,
隐函数F(x,y,z)=z^x-y^z
Fx=lnz*z^x
Fy=-zy^(z-1)
Fz=x*z^(x-1)-lny*y^z
бz/бx=-Fx/Fz=-(lnz*z^x)/(x*z^(x-1)-lny*y^z)
бz/бy=-Fy/Fz=zy^(z-1)/(x*z^(x-1)-lny*y^z)
dz=бz/бxdx+бz/бydy
=-(lnz*z^x)dx/(x*z^(x-1)-lny*y^z)+zy^(z-1)dy/(x*z^(x-1)-lny*y^z)
=(-(lnz*z^x)dx+zy^(z-1)dy)/(x*z^(x-1)-lny*y^z)