Undergrad Calculus in the derivation of Euler-Lagrange equation

[BearY]

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

 

[Charles Link]

Wikipedia does a good write-up. They use the letter $g$ instead of $Y$, but comparing the two, you are correct that it should be a capital $Y$ and $Y'$ in those terms.

 

[BearY]

Wikipedia does a good write-up. They use the letter $g$ instead of $Y$, but comparing the two, you are correct that it should be a capital $Y$ and $Y'$ in those terms.

That Wikipedia page really helped, thanks. I see that $\frac{\partial S}{\partial\alpha}$ when $\alpha = 0$ is the integrand we are looking for. My book didn't bother explaining that directly.