z=f(x/y,xy),求dz

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z=f(x/y,xy),求dz

z=f(x/y,xy),求dz
z=f(x/y,xy),求dz

z=f(x/y,xy),求dz
设 u=x/y,v=xy,则
(其中,ds/dt的比例形式的d是偏导符号;dz等单个形式的d是导数符号,即d)
dz/dx=(dz/du)(du/dx)+(dzdv)(dv/dx)
=(1/y)(dz/du)+y(dz/dv)
dz/dy=(dz/du)(du/dy)+(dz/dv)(dv/dy)
=(-x/y²)(dz/du)+x(dz/dv)
所以
dz=(dz/dx)dx+(dz/dy)dy
=[(1/y)(dz/du)+y(dz/dv)]dx+(-x/y²)(dz/du)+x(dz/dv)dy