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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.

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    $\begingroup$ This expression has no meaning. A partial derivative is with respect to a single variable. $\endgroup$ – Yves Daoust Mar 22 '17 at 12:57
  • $\begingroup$ @YvesDaoust Can i be sure that it has no meaning? For example, the equality $\frac{\partial ((x+y)^2)}{\partial (x+y)}=2x+2y$ holds. What is the difference? $\endgroup$ – Ola Mar 22 '17 at 13:04
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    $\begingroup$ Your second example is actually $\partial t^2/\partial t$ where $t=x+y$. Single variable. See @user1337's answer. $\endgroup$ – Yves Daoust Mar 22 '17 at 13:06
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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.

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