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

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.

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...
$$dy(x,\Delta x):= dy(x)(\Delta x)=y'(x)\Delta x.$$
$$dy(x)$$ 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 $$dy(x,\Delta x):=dy(x)(\Delta x)=y'(x)\Delta x.$$
Normally, we write $$dy$$ instead of $$dy(x,\Delta x)$$.