I'm a freshman in Computer Engineering (but will probably switch to Math very soon) and I've followed some course in Analysis 1 and 2. I found the concept of differentials really interesting, but I've never really grasped the real meaning of the Leibniz notation, such as in:
$$\int {x\ dx} \ \ \ \ \ \ \ \ \ \frac{dy}{dx}$$
In integrals I think of it as a multiplication between a quantity $x$ and another one $dx$. This machine is building small rectangles that will add up to an area. I've studied the integral definitions and I know that there is definitely more behind it (particularly sums), but this is a useful representation that I have in my mind that allow me to conceptualize problems, especially in phisics.
In derivatives I see $\frac{dy}{dx}$ as a division, between something that I want to calculate ($dy$), that is getting smaller and smaller, and an inifnitesimal distance $dx$. But, especially for derivates, I don't really get how is it possible for the ratio of two infinitesimals to output a real number. I know that this is involved with limits, and limits can do these magical things, but it's just not intuitive for me.
Not considering the last doubt, I think that there is much more about the concept of differential that I do not really know. Thus, considering our beautiful math sections we have in our university library, I'm searching for book suggestions about this awesome topic to learn something more about it! Thanks.