Partial derivatives question? I have to find $dw$ if $w=f(u,v,z)$ where $u=x^2+y^2,v=x^2-y^2,z=2xy$.
Now,I know that  $dw=  ( ∂w/ ∂z)*dz  + (∂w/ ∂u)*du +  (∂w/ ∂v)*dv$
The problem is that for example,if I want to find  $∂w/ ∂z$ I don't know how to relate $w$ to $z$? 

(I withdraw my earlier approach.)
We can "solve" the $u$ and $v$ equations for  $x$ and $y$ as
$$2x^2 \ = \ u + v \  \ \ , \ 2y^2  \ = \ u  -  v  \ ,$$
which will allow us to write $ \ z \ $ as 
$$ \ z \ = \ 2xy \ = \ (4x^2y^2)^{1/2} \ =  \ [ \ (u+v)(u-v) \ ]^{1/2} \ = \ [ \ (u^2 -v^2) \ ]^{1/2}, $$
from which we can now extract  
$$dz \ = \ \frac{\partial z}{\partial u} \cdot du \  + \  \frac{\partial z}{\partial v} \cdot dv \ , $$
to replace in your chain-rule equation for $dw$ .
