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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?

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    $\begingroup$ $df(x)$ is defined as $df(x) := f'(x)dx$. So it's not that you're dividing by $dx$ on both sides, it's just that you're recognizing $5$ as the derivative of $y$ from your equation. As for a book on differential forms, how good is your multivariable calculus? If you're pretty confident on it, check out the answer to this question. If not, you're just going to have to wait for now. $\endgroup$ – user137731 Nov 18 '15 at 1:19
  • $\begingroup$ I didn't know there is a definition for df(x) .. Where can I learn that? $\endgroup$ – eugene Nov 18 '15 at 1:20
  • $\begingroup$ Any book on differential forms. For a really brief introduction you could take a look at this pdf. It defines $df$ on page $3$ with partial derivatives, but of course if $f$ is a function of only $1$ variable, then you'd just use the regular derivative. $\endgroup$ – user137731 Nov 18 '15 at 1:20
  • $\begingroup$ The direction you're questioning makes me curious. $\;\;\;$ How did you get $\: dy = 5dx \:$ in the first place? $\;\;\;\;\;\;\;\;$ $\endgroup$ – user57159 Nov 18 '15 at 1:45
  • $\begingroup$ I just gave a simple example I could make up, it's presented in the class (online lecture) $\endgroup$ – eugene Nov 18 '15 at 2:02
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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.

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