*From Taylor's Classical Mechanics, pg. 220:*

> 
> Given: $f(y+\eta (x)\alpha,y'+\eta (x) '\alpha,x)$
> 
> 
> Derivative: $\frac{\partial f(y+\eta (x)\alpha,y'+\eta (x) '\alpha,x)}{\partial \alpha}=\eta \frac{\partial f}{\partial y}+\eta' \frac{\partial f}{\partial y'}$
> 
> 
> 

I guess I can see how the derivative is a summation of 2 terms - it's because there are 2 terms in the argument for the function that depend on $\alpha$. However, I don't see where the $\frac{\partial f}{\partial y}$ and the $\frac{\partial f}{\partial y'}$ come from. I would have just said that the derivative with respect to $\alpha$ is $\eta + \eta '$.

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Update: My main question is just this now: why does the author take the derivative of $f$ w.r.t $y$? He wanted to find the derivative with respect to a parameter that introduced, which is $\alpha$.