Prove equality for multivariate function I am given that $z=f(x^2+y^2)$ and need to show that $y\frac{\partial z}{\partial x}-x\frac{\partial z}{\partial y}=0$.
Listed below is my attempt. Is this correct? I feel that it seems too simplistic.
$\large{
\frac{\partial z}{\partial x} = f'(x^2+y^2) \dot{} 2x\\
\frac{\partial z}{\partial y} = f'(x^2+y^2) \dot{} 2y
}$
Hence,
$$
\begin{align*}
y\frac{\partial z}{\partial x}-x\frac{\partial z}{\partial y} &= f'(x^2+y^2) \dot{} 2x \times y - f'(x^2+y^2) \dot{} 2y \times x
\\&=2xy \times f'(x^2+y^2) -2xy \times f'(x^2+y^2)
\\&=0
\end{align*}
$$

Thanks to the community, it has been verified as correct.
