PDE's - Implicit Differentiation Query

hi,

I am looking at a question in 'advanced engineering mathematics'.

if $\displaystyle z=x^2-y^2$ and $\displaystyle x=r cos \theta$, $\displaystyle y=r sin \theta$

1) Do we say that $\displaystyle z=u(x,y)$ or $\displaystyle z=u(x(r,\theta),y(r,\theta))$ or $\displaystyle z=u(r,\theta)$?

2)I know we can get $\displaystyle \frac{\partial z}{\partial r}=\frac{\partial z}{\partial x} \frac{\partial x}{\partial r}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial r}$and so on for $\displaystyle \frac{\partial z}{\partial \theta}$.

Now lets say we have $\displaystyle z=u(x(r,\theta),y(r,\theta))$

How do I get $\displaystyle \frac{\partial z}{\partial r}$ and $\displaystyle \frac{\partial z}{\partial \theta}$ in terms of $\displaystyle \frac{\partial u}{\partial x}$ and $\displaystyle \frac{\partial u}{\partial y}$?

Do I differentiate both sides of the equation?

Thanks