Mechanics derivation that I don't understand

I am reading the section of method of calculus of variations from Goldstein, where he tries to find a curve for which given line integral has a stationary value.

After some steps into the derivation we have to first define $$J(\alpha) = \int_{x_1}^{x_2}f(y(x,\alpha),\dot{y}(x,\alpha),x)\,dx$$ The condition for obtaining a stationary value is the following: $$\left(\frac{dJ}{d\alpha}\right)_{\alpha = 0} = 0$$ So taking the derivative we get the following: $$\frac{dJ}{d\alpha} = \int_{x_1}^{x_2}\left(\frac{\partial f}{\partial y}\frac{\partial y}{\partial \alpha} + \frac{\partial f}{\partial \dot{y}}\frac{\partial\dot{y}}{\partial \alpha}\right) dx \,.$$ Considering the second of these integrals, we get $$\int_{x_1}^{x_2} \frac{\partial f}{\partial \dot{y}}\frac{\partial \dot{y}}{\partial \alpha} dx = \int_{x_1}^{x_2} \frac{\partial f}{\partial \dot{y}}\frac{\partial^2 y}{\partial x\,\partial \alpha} \, dx$$ (I don't understand this step)

Is it true that $$\partial \left(\frac{dy}{dx}\right) = \frac{\partial x}{\partial y} \,?$$

Another thing is once we integrate by parts the above we get $$\int_{x_1}^{x_2} \frac{\partial f}{\partial \dot{y}}\frac{\partial^2 y}{\partial x\partial \alpha}dx = \frac{\partial f}{\partial \dot{y}}\left.\frac{\partial y}{\partial \alpha}\right|_{x_1}^{x_2} - \int_{x_1}^{x_2}\frac{d}{dx}\left(\frac{\partial f}{\partial \dot{y}}\right)\frac{\partial y}{\partial \alpha}$$ I don't understand how did we get this by integration by parts as derivatives and partials aren't really the same thing. If someone could explain that would be great.

When you consider de second of the integrals, we can define: $$\dot{y}\equiv\frac{\partial y}{\partial x}\Rightarrow \frac{\partial \dot{y}}{\partial \alpha}=\frac{\partial }{\partial \alpha}\left(\frac{\partial y}{\partial x}\right)=\frac{\partial ^2y}{\partial \alpha\partial x}$$ The notation $\partial \left(\displaystyle\frac{dy}{dx}\right)=\displaystyle\frac{\partial x}{\partial y}$ would not be correct, because the function is $y=y(x,\alpha)$, a function of two variables, and $x$ is one of the variables, so you cannot exchange their position on the derivative. Finally, when you integrate by parts, although partial derivatives and total derivatives are not the same thing, you can integrate "partially" leaving the other variables as constants, so you can integrate a second order partial derivative, getting a first order partial derivative, as if it were an ordinary derivative. Hope it helps.