Why is it legitimate to perform multiplication with differentials $dx$?
For instance, from the statement $dy = 5dx$ one derives $\frac{dy}{dx} = 5$.
I learned $\frac{dy}{dx}$ as a notation to mean the limit of the rate of change.
In this MIT OpenCourseWare video, the professor states that $dx$ is not a number but doesn't define what it is precisely.
Is there a book/online doc that talks about why it is legitimate to manuplate $dx$ as if it were a value?
A lot of the manipulations people do with differentials can be understood by thinking of $dx$ as a differential form. Unfortunately, it's hard to find a good introduction to differential forms that doesn't assume the reader is already very comfortable with calculus. You might try David Bachman's A Geometric Approach to Differential Forms.