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## Homework Statement

I need to prove that [itex]x\frac{ \partial^2z}{ \partial x^2} + y\frac{\partial^2z}{\partial y\partial x} = 2\frac{\partial z}{\partial x}[/itex]

## Homework Equations

[itex] z = \frac{x^2y^2}{x+y} [/itex]

## The Attempt at a Solution

I actually did it the long way and I got the right answer but here is my teacher's solution :

[itex] z = \frac{x^2y^2}{x+y} [/itex]

[itex]\Rightarrow x\frac{ \partial z}{ \partial x} + y\frac{ \partial z}{ \partial x} = 3\frac{ \partial z}{ \partial x} [/itex]

[itex]\Rightarrow \frac{ \partial z}{ \partial x} +x\frac{ \partial^2z}{ \partial x^2} + y\frac{ \partial z}{ \partial x} = 3\frac{ \partial z}{ \partial x} [/itex]

Answer follows.

To be honest, I have absolutely no idea about what technique he actually uses there. Is there any "rule" or "trick" that I am not aware of here?