Chain rule - Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y} $ if $z=pq+qw$

Question

Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y} $ if

$z=pq+qw$

$p=2x-y$, $q=x-2y$ and $w=-2x-2y$

Is $\frac{\partial z}{\partial x}$ equals to:

$\frac{\partial z}{\partial p} \frac{\partial p}{\partial x}+ \frac{\partial z}{\partial q} \frac{\partial q}{\partial x}+ \frac{\partial z}{\partial q} \frac{\partial q}{\partial x}+ \frac{\partial z}{\partial w} \frac{\partial w}{\partial x}$

Or $\frac{\partial z}{\partial p} \frac{\partial p}{\partial x}+ \frac{\partial z}{\partial q} \frac{\partial q}{\partial x}+ \frac{\partial z}{\partial w} \frac{\partial w}{\partial x}$

I can complete the solution, I need only the correct rule. Thanks

Answer 1

$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial p} \frac{\partial p}{\partial x}+ \frac{\partial z}{\partial q} \frac{\partial q}{\partial x}+ \frac{\partial z}{\partial w} \frac{\partial w}{\partial x}=$, then

$\frac{\partial z}{\partial x}=p+w$ and

$\frac{\partial z}{\partial y}=-3q-2p-2w$

Answer 2

$$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial p} \frac{\partial p}{\partial x}+ \frac{\partial z}{\partial q} \frac{\partial q}{\partial x}+ \frac{\partial z}{\partial w} \frac{\partial w}{\partial x}$$