Link to the Basic definition's image
What is the theory of the transformation from $f'(x)$ into $\frac{{\partial f}}{{\partial x}}(x,y)$ ? (I don't mean just replace a by y) Could anybody explain shortly and give me some reference? Your help is appreciated.
They're just replacing $y$ by $a$ to emphasise the fact that $y$ is being considered as a constant in forming the partial derivative with respect to $x$; then, when the derivative has been formed, they substitute $y$ for it again. There isn't more to it than that; if you understand partial derivatives without this detour, there's no need to apply it.