How to find $\frac{\partial (xy)}{\partial (x+y)}$

Question

Is there a function $f(x,y)$ such that $f(x,y)=\frac{\partial (xy)}{\partial (x+y)}$? After some subtitution i found that $f(x,y)=x+(x-1)\frac{\partial (x)}{\partial (x+y)}$, but that was as far as far as i got. Any advice is appriciated.

Answer 1

The sum and product of two numbers are completely independent of each other. That is the equations $$x+y=a \\xy=b $$

are solvable for all choices of $a$ and $b$. Thus $xy$ is not a even a function of $x+y$, and consequently cannot be differentiated with respect to it.