So, our starting equation is

When computing , we are considering z to be a function of x, i.e., z(x). So, any time we have a term containing z multiplied with a term containing x, we must use the product rule.

So now we differentiate:

Looking at these two partials:

and

Thus:

At this stage, you can simplify and solve for .

I'll leave the other partial for you to solve using the same technique.