I am assuming you simply need to verify the second equation given the first?

z = f(x^2 - y^2)

So

zx = f'(x^2 - y^2)*(2x)

zy = f'(x^2 - y^2)*(-2y)

So:

y(zx) + x(zy) =?0

y*f'(x^2 - y^2)*(2x) + x*f'(x^2 - y^2)*(-2y)

= 2xy*f'(x^2 - y^2) - 2xy*f'(x^2 - y^2) = 0 (Check!)

-Dan