We can use implicit differentiation letting y be a constant.
Differentiate both sides partially with resepct to x:
The chain rule has been used on the left hand side to differentiate z^2 and the product rule has been used on the right to differentiate the product.
Now re-arrange the result to make the subject.
Same idea when getting the partial derivative with respect to y.
Note that you can re-arrange the given rule to make z the subject and then take the partial derivative with respect to x in the usual way.