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Euler-Lagrange equation

$$\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial y'} = 0$$

Can also be written as

$$f'_y-f''_{xy'}-f''_{yy'}y'-f''_{y'y'}y''=0$$

In my book it is provided as a self-evident fact. It is not evident to me at all. Please help me with the derivation, I have no idea where to even start

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$y=y(x)$, $f=f(x,y,y')$. Then $$\frac{d}{dx}\frac{\partial f}{\partial y'}=\frac{d}{dx}y'_{y'}(x,y,y')= f''_{xy'}+f''_{yy'}y'+f''_{y'y'}y''$$

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