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What is the correct way of writing derivatives in notation form? Should the 'd' be upright or italic, that is $\mathrm d$ or $d\,$? As an example, should we write: $$ \frac {\mathrm dy}{\mathrm dx}\qquad\text{or}\qquad \frac {dy}{dx} \, \text ?$$

I feel the first method (upright 'd') is correct, since italic letters are generally used for variables, but most people (and Wikipedia) use the italic 'd'. Which is actually more correct?

Also, how do we use this notation on notebooks (writing by hand), while avoiding potential confusion in a variable $d$ and the differential symbol? Since mostly we write variables upright on notebooks, so $\displaystyle \mathrm{\frac{dd}{dt}}$ in a notebook would be potentially confusing. (Of course, we could avoid a variable 'd', but still, what would be another way out of this?)

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    $\begingroup$ I can't imagine a situation where it would make a difference. It should be clear from context regardless of if the letters are italicized or not. $\endgroup$ – Sean English Mar 27 '16 at 15:30
  • $\begingroup$ The esdiff package does it by default (including for partial derivatives) and has a simple syntax, useful for higher order derivatives. Often people use italic letters by pure laziness. $\endgroup$ – Bernard Mar 27 '16 at 15:30
  • $\begingroup$ The vast majority of mathematics journals and books use italic $d$, I think. But there exists an international standard for notation in technical and engineering fields that specifies the upright $\mathrm{d}$ for this purpose, and similarly for $\mathrm{i}=\sqrt{-1}$ and $\mathrm{e}=2.71828…$ (sorry, I don't have the reference on hand). $\endgroup$ – Harald Hanche-Olsen Mar 27 '16 at 15:33
  • $\begingroup$ Not only for technical and engineering fields. It's been used in French mathematics for many years. $\endgroup$ – Bernard Mar 27 '16 at 15:47
  • $\begingroup$ Sooner or later you'll have to pay for using the upright ${\rm d}$, because it's ISO-certified. $\endgroup$ – Christian Blatter Mar 27 '16 at 15:50
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It is most common to use italic $d$, probably at least in part because that's the default LaTeX does. I have adopted a habit of handwriting with cursive to help distinguish symbols.

I write mostly in print, but I write $dx$ cursive and conjoined, which emphasizes that the two letters form one mathematical symbol. I write the differential operator as a cursive $d$ not conjoined with the following text.

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  • $\begingroup$ TeX was much later (early 1980s) than the prevalence of italics in mathematics. $\endgroup$ – ForgotALot Mar 27 '16 at 16:12
  • $\begingroup$ I liked the cursive idea for notebooks! Just one more thing, what about on computers (like on websites such as Math.SE)? Do you know about the standard there? $\endgroup$ – FreezingFire Mar 27 '16 at 16:51
  • $\begingroup$ @ForgotALot yes, I only mean that it's current prevalence is in part because its been built into the system $\endgroup$ – Stella Biderman Mar 27 '16 at 18:24
  • $\begingroup$ @FreezingFire on websites such as MSE we are writing in LaTeX. Other places with built in equation editors, I just use the default, because I don't use built in equation editors enough it to be a big deal $\endgroup$ – Stella Biderman Mar 27 '16 at 18:26

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