Deriving Relations for Partial Derivatives in a System of Four Variables

[fluidistic]

Homework Statement

Given 4 state variables x, y, z and w such that $F(x,y,z)=0$ and w depends on 2 of the other variables, show the following relations:
1)$\left ( \frac{\partial x }{\partial y } \right ) _z = \frac{1}{\left ( \frac{\partial y }{\partial x } \right ) _z}$
2)$\left ( \frac{\partial x }{\partial y } \right ) _z \left ( \frac{\partial y }{\partial z } \right ) _x \left ( \frac{\partial z }{\partial x } \right ) _y=-1$
3)$\left ( \frac{\partial x }{\partial w } \right ) _z=\left ( \frac{\partial x }{\partial y } \right ) _z\left ( \frac{\partial y }{\partial w } \right ) _z$
4)$\left ( \frac{\partial x }{\partial y } \right ) _z=\left ( \frac{\partial x }{\partial y } \right ) _w+\left ( \frac{\partial x }{\partial w } \right ) _y \left ( \frac{\partial w }{\partial y } \right ) _z$

Homework Equations

Hints: for 1) and 2) think about x as x(y,z) and then y=y(x,z)
For 3) choose x=x(x,z)
For 4) choose x=(y,w)

The Attempt at a Solution

Stuck on 1). I'd be tempted to consider differentials like numbers and that way 1) would be instantly "proven". However I do not see how to use the tips provided.

 

[sunjin09]

I assume (dx/dy)_z means derivative of x w.r.t. y when z is fixed.
since F=0, dF=F_x dx+F_y dy+F_z dz=0. When z is fixed, dz=0, so F_x dx=-F_y dy, etc.

 

[fluidistic]

I assume (dx/dy)_z means derivative of x w.r.t. y when z is fixed.
since F=0, dF=F_x dx+F_y dy+F_z dz=0. When z is fixed, dz=0, so F_x dx=-F_y dy, etc.

Thank you very much for this huge tip. Will be working on that problem and post if I'm stuck.