I have the variable $x,y,z$ possibly depending on each other i.e. on a smooth manifold. Using the theory of differential forms I can derive $\left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial y}{\partial z}\right)_x=-1$ by \begin{align}dx\wedge dz=\left(\frac{\partial x}{\partial y}\right)_z dy\wedge dz=\left(\frac{\partial x}{\partial y}\right)_z\left(\frac{\partial z}{\partial x}\right)_y dy\wedge dx =\left(\frac{\partial x}{\partial y}\right)_z\left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial y}{\partial z}\right)_x dz\wedge dx\end{align}
I don't understand the formal mathematical meaning of the subscripts. They are indicating which variable is held constant (why do I need this information?).
Now I should proof the following for the variables $x,y,z,w$:\begin{align} \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\end{align}
Not only that I don't know how to prove it, I don't even understand the different meaning of $\left(\frac{\partial x}{\partial y}\right)_z$ and $\left(\frac{\partial x}{\partial y}\right)_w$. Any advice regarding the concrete problem and some reading advice on the general topic are very welcome . I tried Schutz's geometrical methods of mathematical physics, but I'd like something with a focus on purer mathematics.