# Implicit differentiation in multivariable calculus

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.