Consider $R=\{(x,y) \in \Bbb R^2 : x>0, y>0\}$ Now on $R$ consider the transformation $(x,y) \mapsto (z \sqrt{w}, w)$ i.e. $x=z\sqrt{w}$ and $y=w$ for $z>0,w>0$ . Write $\frac{\partial}{\partial x}$ in terms of $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial w}$
My attempt:
By chain rule, $$\frac{\partial}{\partial x}=\frac{\partial z}{\partial x}\frac{\partial}{\partial z}+\frac{\partial w}{\partial x}\frac{\partial}{\partial w}$$ Now, $z=\frac{x}{\sqrt y}$ and thus $\frac{\partial z}{\partial x}=\frac{1}{\sqrt{y}}$
And $\frac{\partial w}{\partial x}=\frac{\partial y}{\partial x}=0$
So $\frac{\partial }{\partial x}=\frac{1}{\sqrt{w}} \frac{\partial }{\partial z}$
Am I correct? I am a just a bit confused because we got $\frac{\partial w}{\partial x}=0$ , although we have $x=z\sqrt{w}$ and hence, $w=\frac{x^2}{z^2}$ So we have $w$ explicitly in terms of x, but still the partial derivative is zero!
I know this might be trivial but I am in high school and completely new to calculus.