Euler-Lagrange Variational Calculus extreme condition for generic functional

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I need to do the following :

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I saw for a simpler case, but I didn't understand the passage indicated by an arrow in the image below .

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Why did the top integral become the bottom subtraction?

Now, I need to generalize to the case f(y, y', y'', x) , but I don't even understand the case for f(y, y', x).