Lecture 15: Antiderivatives | Video Lectures | Single Variable Calculus | Mathematics | MIT OpenCourseWare

I am going through the video above. There is some notation being introduced:

It is my understanding right now that "*dx" is only there because it is denoting sin(x) as being a derivative of some equation and that it does not at all mean sin(x) times dx. Am I wrong on that?

Also it is introduced that:

So I have studied I'm not sure yet if this is related however I find it very confusing that in the case above dx is on the top part of a fraction? I'm not sure what to make out of that.

Thanks for any responses...