Given the function $$g(x,y)=f(x^n+y^n,x^n-y^n)$$ I have to show that $$\frac{1}{nx^{n-1}}\cdot\frac{\partial g}{\partial x}(x,y)+\frac{1}{ny^{n-1}}\cdot\frac{\partial g}{\partial y}(x,y)=0$$
If $u=x^n+y^n$ and $v=x^n-y^n$, then $$\frac{\partial u}{\partial x}=nx^{n-1},\quad\frac{\partial u}{\partial y}=ny^{n-1}$$ and $$\frac{\partial v}{\partial x}=nx^{n-1},\quad\frac{\partial v}{\partial y}=-ny^{n-1}$$ Then $$\frac{\partial g}{\partial x}=\frac{\partial f}{\partial u}\cdot\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\cdot\frac{\partial v}{\partial x}$$ $$\frac{\partial g}{\partial x}=nx^{n-1}\cdot\left(\frac{\partial f}{\partial u}+\frac{\partial f}{\partial v}\right)$$ In the same way $$\frac{\partial g}{\partial y}=ny^{n-1}\cdot\left(\frac{\partial f}{\partial u}-\frac{\partial f}{\partial v}\right)$$ Therefore, $$\frac{1}{nx^{n-1}}\cdot\frac{\partial g}{\partial x}(x,y)+\frac{1}{ny^{n-1}}\cdot\frac{\partial g}{\partial y}(x,y)=\frac{\partial f}{\partial u}+\frac{\partial f}{\partial v}+\frac{\partial f}{\partial u}-\frac{\partial f}{\partial v}=2\frac{\partial f}{\partial u}\neq0$$ What is wrong?