A constrained differential probelm

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SUMMARY

The discussion centers on the constrained differential problem defined by the function $$w(x,y,z)=zxe^y+xe^z+ye^z$$ with the constraint equation $$x^2y+y^2x=1$$. The differential derived is $$dy=-\frac{2xy+y^2}{2xy+x^2}$$. A key point of contention is the substitution of $$z=0$$ when calculating $$\frac{\partial w}{\partial x}$$, which some participants find unclear despite understanding that $$dz=0$$. The discussion seeks clarification on the rationale behind this substitution.

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Leo Liu
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Homework Statement
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Define that $$w(x,y,z)=zxe^y+xe^z+ye^z$$
1614266061848.png

So the constraint equation is ##x^2y+y^2x=1##. And its differential is ##dy=-\frac{2xy+y^2}{2xy+x^2}##.

However, the solution plugs in ##z=0## when computing ##\frac{\partial w}{\partial x}## as shown in the screenshot below. While I understand that ##dz=0##, I can't see why ##z=0##. Could anyone explain?
1614266672731.png
 
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Can you show us the first parts of the problem?
 
Office_Shredder said:
Can you show us the first parts of the problem?
Sure, Here it is.
1614301330031.png
 

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