Simplifying a chain rule in a model.

Simplifying a chain rule in a model.

I was reading a paper (Taylor and Frank, 1996). There is a chain rule of a
function W(y,z) thus:
dW/dx = partial(dW/dy)*dy/dx + partial(dW/dz)*dz/dx
I get this bit but then the two derivatives dy/dx anf dz/dx are both
expressed interms of regressions i.e Beta(zx) and Beta(yx). I can
understand that notation i think. Then the authors say they divide both
sides by Beta(yx) and reach the following:
Delta(W) = partial(dW/dy) + partial(dW/dz)*R
Where R= Cov(zx)/Cov(y, x)
And this is where i get lost. How does the left hand side simplify? Why is
R's numerator cov(zx) and not cov(z,x)? what does the lack of a comma
mean?
Hope this is clear, If not please say and i will format the equations
properly.
Thank you for your time to help this poor neophyte.
HF