At first, I think my calculus knowledge is already sufficient. However, after I encounter many much usage of calculus that seems weird to me, I must rethink about my standing point. I just want to ask if anyone know any good, outside reading book on calculus that explain the meaning of many calculus elements and how it works.

For example, I'm very confuse on the fact that physics uses calculus such as, $\displaystyle dA$ as a very small infinitesimal area, where integral of that produce the total Area. Therefore, in a sense, $\displaystyle dA$ is often treated like a variable. However, in calculus, there's such thing such as $\displaystyle d/dx( )$ or $\displaystyle D$ operator, and even Half-derivative that seems confusing in physical context. This operator idea seems to contradict with my picture of "infinitesimal variable". Another thing that I am confused is the idea of high order derivative, if I think in "operator" sense, it is like $\displaystyle d/dx(dy/dx)$ for 2nd order derivative. However, I do not get a meaning about it when apply on physical context, Let's say distance derived respect to time to the 2nd order: $\displaystyle d(dr)/dt^2$, it is like $\displaystyle d(dr)/dt^2$... Infinitesimal of the Infinitesimal of distance? or Infinitesimal changes of the Infinitesimal changes of distance?

So if there's some books that really deeply explain about calculus' elements, I'd appreciate very much.