Q:

Problem in understanding a concept of chain rule.

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.

A:

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 $F$.

