Partial derivative - why is it count like this?

[player1_1_1]

Homework Statement

I have John R Taylor "Classical mechanics" part 1, and I have an integral:
$$\int\limits^{x_2}_{x_1}f\left(y+\alpha\eta,y^{\prime}+\alpha\eta^{\prime},x\right)\mbox{d}x$$
and here is count derivative of underintegral function in $$\alpha$$
$$\frac{\partial f\left(y+\alpha\eta,y^{\prime}+\alpha\eta^{\prime},x\right)}{\partial\alpha}=\eta\frac{\partial f}{\partial y}+\eta^{\prime}\frac{\partial f}{\partial y^{\prime}}$$
why there is suddenly $$\frac{\partial f}{\partial y^{\prime}}$$ and $$\frac{\partial f}{\partial y}$$, while this derivative is by $$\alpha$$ - why not only $$\eta,\eta^{\prime}$$?

Homework Equations

I was thinking about function composition derivative, but it didnt helped me.

The Attempt at a Solution

Nothing, I couldn't do anything with this, I don't know why this is count like this, please help;] thanks!

 

[phsopher]

$$f=f(y,y',x)$$

Now if you replace $$y=\bar{y}+\alpha\eta$$ then y becomes a function of alpha, which means that you have to use the chain rule.

Try choosing an explicit non-linear expression for f if that makes it more clear. For example f(y,y',x) = y² + y'²