How to find $\frac{∂f}{∂y}$ for Functions of the form $f(x,y,z(y))$? This is the first time I'm using math.stackexchange. Please excuse me and correct me if I'm not doing things in the right format.
So my question is this: given a function of the form $f(x,y,z(y))$, and suppose we want to find $\frac{∂}{∂y} f(x,y,z(y))$. Then by the Chain Rule, we would have something like this
$$\frac{∂f}{∂y} = \frac{∂f}{∂y} + \frac{∂f}{∂z}\frac{dz}{dy}.$$
But this notation is really confusing since the two $\frac{∂f}{∂y}$ do not mean the same thing. Am I doing this correctly, or is there any better notation to clarify this expression? I would be much appreciated if someone could shed some light on me.

Don't confuse the function $f(x,y,z)$ and the function $F(x,y)=f(x,y,z(y))$. 
The first is a function of three variables $x,y,z$ while the second is a function of two variables $x,y$. 
People familiar with multi variables calculus loosely use a common symbol for both without making mistake. For one not yet familiar with these symbolism it is suggested to use different symbols : 
$$\frac{∂F}{∂y}=\frac{∂f}{∂y}+\frac{∂f}{∂z}\:\frac{dz}{dy}$$
