Fixed Variables in Partial Derivatives

[transmini]

My physics book is showing an example of why it matters "what variable you hold fixed" when taking the partial derivative. So it asks to show that
$(\frac{\partial{w}}{\partial{x}})_{y} \neq (\frac{\partial{w}}{\partial{x}})_z$
where $w=xy$ and $x=yz$ and the subscripts are what variable is held fixed.

I'm just not sure what it means by holding a variable "fixed" since all other variables except the two in question are treated as constants, so why would it matter whether $w$ is a function of $(x, y)$ or a function of $(x, z)$
I mean I see that the partial derivatives don't match up, but don't really see why the variables make a difference since they are the same function.

 

[Orodruin]

The condition $x=yz$ describes a surface in three dimensions. Now, you can use different sets of variables to parametrise this surface. One choice is to use x and y and another to use x and z. Varying x in the first parametrisation is generally going to give you a displacement in the surface that will be different from the one you obtain if you vary x in the same way in the second parametrisation. The "keeping these variables fixed" essentially tells you what parametrisation is used and therefore exactly what variations should be considered.