What does $(dy)(x, \Delta x)$ mean?

Question

I am reading ordinary differential equations from a book which says

Hence, $dy = f'(x) \Delta x$, we call $dy$ the differential of $y$. As the differential $dy$ is a function of two independent variables $x$ and $\Delta x$, we indicate this dependency by $(dy)(x, \Delta x)$.

Excuse me but what the hell is this? It goes even further by saying

The differential of $y$, written as $dy$ (or $df$) is defined by $$(dy)(x, \Delta x) = f'(x) \Delta x$$

And now I am completely confused. Do we define $dy$ twice? Or what is $(dy)(x, \Delta x)$? Is it a sort of relation as it says "we write a dependency in this way", so it is a relation in terms of set theory? What is it exactly?

Why do $dy$ and $(dy)(x, \Delta x)$ have the same definitions? Can we say that $dy = (dy)(x, \Delta x)$ then? If so, what is the point of this mess? Thank you if you read up to this point and I will be more thankful if you help me to understand what I am missing.

Answer 1

Say $y=e^{2x}$. Then we write $dy = 2 e^{2x}\;dx$. This exhibits $dy$ as a function of two variables, namely $x$ and $dx$. Perhaps this author wrote $\Delta x$ in place of $dx$, hoping that this would reduce confusion...

Answer 2

$$dy(x,\Delta x):= dy(x)(\Delta x)=y'(x)\Delta x.$$

$dy(x)$ is a linear application : it's the linear approximation of $y$ in a neighborhood of $x$, i.e. $dy(x):\mathbb R\to \mathbb R$ if defined by $$dy(x)(h):= y'(x)h.$$ But $h=x+h-x=:\Delta x$, so you have to see $\Delta x$ as a "variable" (namely as the distance from $x$), and we denote $$dy(x,\Delta x):=dy(x)(\Delta x)=y'(x)\Delta x.$$

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Normally, we write $dy$ instead of $dy(x,\Delta x)$.