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What I don't understand is the disconnect between the reasoning for implcit derivation in single variable and multivaraible calc.

in single variable calc they define say $y = f(x)$ for a small interval which is not tradionally a function.

So we get $f(x,y(x))$

How would I group the variables?

Say I would like to find:

$\frac{\eth x}{\eth z}$ given the function $F(x,y,z,w)=0$ whilst holding $w$ fixed?

They solve this by differentiating the equation with respect to $z$, regarding $x$ and $y$ as functions of $z$ and $w$, and holding $w$ fixed.

My reasoning thus far is that since $x$ is the dependent varaible we could write x=(y,z,w)

as we would previously for $f(x,y)$

Why is this not right?

Instead they say that

differentiating the equation with respect to $z$, regarding $x$ and $y$ as functions of $z$ and $w$, and holding $w$ fixed.

meaning that

$x = x(z,w)$

$y = y(z,w)$

I could understand why they would this for $x$ but why would they treat $y$ also as an dependet varaible? What implies this? When looking for $\frac{\eth x}{\eth z}$

where

$F(x,y,z,w)=0$

and $w$ is fixed.

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