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In the derivation of Euler-Lagrange equation, when differentiating S with respect to α, there is a step:

$$\frac{\partial f(Y,Y',x)}{\partial\alpha}=\frac{\partial f}{\partial y}\frac{\partial y}{\partial\alpha}+\frac{\partial f}{\partial y'}\frac{\partial y'}{\partial\alpha}$$

Where $$ Y = y(x)+\alphaη(x)$$

My puny math knowledge can't tell me 2 things:

1.why is it ##\frac{\partial y}{\partial\alpha}## instead of ##\frac{\partial Y}{\partial\alpha}##? Isn't the second one equal to η? Why is the first one equal to η? Did I skip something?

2. Where does the plus sign come from? I learned partial derivative before but I cannot recall anything like this. I have a feeling this is the result of forgetting something completely

Edit: NM the second one I was stupid It's just chain rule

$$\frac{\partial f(Y,Y',x)}{\partial\alpha}=\frac{\partial f}{\partial y}\frac{\partial y}{\partial\alpha}+\frac{\partial f}{\partial y'}\frac{\partial y'}{\partial\alpha}$$

Where $$ Y = y(x)+\alphaη(x)$$

My puny math knowledge can't tell me 2 things:

1.why is it ##\frac{\partial y}{\partial\alpha}## instead of ##\frac{\partial Y}{\partial\alpha}##? Isn't the second one equal to η? Why is the first one equal to η? Did I skip something?

2. Where does the plus sign come from? I learned partial derivative before but I cannot recall anything like this. I have a feeling this is the result of forgetting something completely

Edit: NM the second one I was stupid It's just chain rule

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