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1. The problem statement, all variables and given/known data

Can the partial derivative of a function depend depend on the form it is in?

Say, z = f(x,y), and y=g(x,w). If I take

[tex]\frac{\partial z}{\partial y} [/tex]

then I get

[tex]\frac{\partial f(x,y)}{\partial y}[/tex]

which is not necessarily 0. But [itex]\frac{\partial z}{\partial y} [/itex] is also equal to

[tex]\frac{\partial f(x,g(x,w))}{\partial y}[/tex]

which is identically 0. This is DRIVING ME OUT OF MY MIND.

Also, say we have z = f(x,y) = x^2+y^2+y and we also know that x=y. Then z also equals g(x,y) = x^2+y^2+x.

Thus, we get

[tex]2 y +1 = \frac{\partial f(x,y)}{\partial y} = \frac{\partial z}{\partial y} = \frac{\partial g(x,y)}{\partial y} = 2y[/tex]

which is absurd. What is wrong with my logic?

All of these examples come from thermodynamics.

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Partial derivatives

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