Suppose $f(x,y,z,w)=0$ be a differentiable equation of $x,y,z$ & $w$ ,where $z,w$ are differentiable functions of $x,y$.Then my book suggests that if we apply chain rule we have $f_x+f_z.z_x+f_w.w_x=0$.But I have a confusion here.Suppose $f(x,y,z,w)=x^2+2y+z^3+w^2$ where $z=x+y$ and $w=x-y$.Then according to my book $f_x=2x$ but as far as I know $f_x=2x+3z^2.z_x+2w.w_x \neq 2x$ in general.Isn't it?If the answer is "yes" then how should I mention the term $2x$.Is it also a partial of $x$ or does it signify any other meaning.Please help me in understanding this fact.
Thank you in advance.
The usual notation for partial derivatives leads to all sorts of confusion, so you are not alone in tripping up here. I hope this unpicking of what the formulas mean helps.
To simplify what I write I assume all the functions are defined everywhere.
So we have a function of four variables $f:\mathbb{R}^4\to\mathbb{R}$, and two functions of two variables $z,w:\mathbb{R}^2\to\mathbb{R}$. From these we construct a new function, $F:\mathbb{R}^2\to\mathbb{R}$ by $F:(x,y)\mapsto f(x,y,z(x,y),w(x,y))$.
The chain rule now gives $$\begin{matrix} F_1 &=& f_1.1+f_2.0+f_3.z_1+f_4.w_1\\ F_2 &=& f_1.0+f_2.1+f_3.z_2+f_4.w_2\\ \end{matrix} $$ where I have written $\phi_i$ to denote the $i$-th partial derivative of a $\phi$.
Note that if you insist on replacing the $1$'s and $2$s by $x$'s and $y$'s, confusion can arise because the notation $\phi_x$ means different things on the left hand side and on the right hand side.
With $f(x,y,z,w)=x^2+2y+z^3+w^2$, we have that $$f_1=2x, \text{ whereas } F_1= 2x + 3z^2 z_1 + 2w w_1$$ which in the standard (but confusing) notation is $$f_x=2x, \text{ whereas } F_x= 2x + 3z^2 z_x + 2w w_x.$$
Now what about the hypothesis "Suppose $f(x,y,z,w)=0$ be a differentiable equation of $x,y,z,w$ ,where $z,w$ are differentiable functions of $x,y$"?
This is again just too compressed if we want to avoid confusion. If $f$ were indeed $0$ then all its partial derivatives would be $0$ and everything would vanish. What the code means is that when we substitute in $z(x,y)$ for $z$ and $w(x,y)$ for $w$ then we get $0$. That is, the hypothesis really is that the function $F=0$, and so what we have is that $F_1=F_2=0$.
By the way ignore what @MyGlasses and @Upax say, they are confusing $f$ and $F$.