Can someone please explain why , where y is a function of x, is necessarily zero? Is it because holding x in place forces y' to never change?
Using the chain rule we get d/dy (y'(x)) = dy'(x)/dy * dy/dy = 0 * 1 = 0. I'm not sure if this answer suffices or not to answer your question, but the chain rule is always a good tool when you have these kinds of derivatives.
The aim of my first post at arousing the controversy because the wording of the question was ambiguous.
Introducing the partial derivative of (dy/dx) relatively to y suppose that dy/dx might be function of y, i.e.: dy/dx = f(x,y). There are many different manner to write f(x,y) and all are equal.
For example, in the case y(x)=x²
f(x,y) = 2x, or f(x,y)=2sqrt(y), or f(x,y)=x+sqrt(y), or f(x,y)=2y/x, or etc.
The partial derivative relatively of y is different for each one of these functions f(x,y). So, without more specification in the wording, the question is ambiguous.
Among many possibilities, if we say : y(x) relates y to x as well as the inverse function x(y), then dy/dx is function of x which is function of y. Hence dy/dx is function of y. On this point of view, the formula for d(dy/dx)/dy is shown in attachment.