Math CentralQuandaries & Queries


Question from Shafira, a student:

In all math textbooks, it is written that d/dx ( d/dx) (y)= d2y/dx2. Why do they write it as d2y/dx2, not as d2y/d2x2?


The "/" in "d/dx" is not really a fraction, but it behaves enough like one (eg, in the chain rule) that it is convenient to pretend it is notationally. We still have to be aware that it is not division and that we can't do everything that we could it if were.

In the same way, in the alternative "operator notation" notation Dy [differentiating with respect to an agreed-upon but unstated variable] the D acts a little bit like a multiplication, so we can write things like $(D^2+ 2D+3)y$ meaning $y" + 2y' + 3y.$ This is useful in differential equations. But we must be very careful here; for instance, $Dy/Dz$ is not equal to $y/z.$

But in the symbol "dx" the d is not even an operator and we cannot usefully pretend that it is being multiplied by the x. The differential "dx" must be treated as an unbreakable unit; so we square it, not its component letters. In the same way, we never write $sin^2(x)$
as $s^2i^2n^2(x)$. (If we did, noting that $ i^2=-1$ we could simplify to $-sn^2(x),$ which is completely silly.)

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