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

When we take the partial derivative of z with respect to y we take an "ordinary" derivative, but assume that y is constant. Thus:

(del)z/(del)x = f'(x^2 + y^2) * (2x)

where (del)z/(del)x represents the partial derivative and f'(u) represents the derivative of f(u) with respect to its argument. This is nothing more than the chain rule where we have set y equal to a constant.

Similarly:

(del)z/(del)y = f'(x^2 + y^2) * (2y)

So:

y * (del)z/(del) - x * (del)z/(del)y

= 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

as advertised.

-Dan