What should I study to understand how to operate with differentials such as "dx"? I study Physics and I have study real analysis little time ago. It is common practice in physics to operate with differentials in a very informal way. 
 For example : When solving differential equations we multiply and cancel out dx's all day long, sometimes we hear that we just need to think about them as little delta x.
My question is: What should I study, to understand formally the nature of these mathematical object "dx" "dy" and how to operate with them, it makes me preety confused when , for example, we try to solve for the lenght of a curve and some dx² shows up.
I remember when in some physical arguments we just vanish higher powers of dx saying that " if dx is small dx² is even smaller " haha, i Know its funny, i want to get rid of this informal understanding and have a solid background. 
Thank you guys in advance.
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You are asking about differential forms. Check out Barrett O'Neil's Differential Geometry, introduces forms quite nicely in the first chapter. 
I also recommend these lectures on Differential Forms https://www.youtube.com/watch?v=Nh5XFX0iKgE by Professor Theodore Shifrin. Not only do his lectures deal with the setup of forms, but he proves the Generalized Stokes Theorem with them. Some background with linear algebra is helpful to fully understand the videos: Vector Space, Basis, Span, Linear Independence, Linear Transformations/Maps, Determinants, etc. 
