In school I was taught that we use $\frac{du}{dx}$ as a notation for the first derivative of a function $u(x)$. I was also told that we could use the $d$ just like any variable.
After some time we were given the notation for the second derivative and it was explained as follows:
$$
\frac{d(\frac{du}{dx})}{dx} = \frac{d^2 u}{dx^2}
$$
What I do not get here is, if we can use the $d$ as any variable, I would get the following result:
$$
\frac{d(\frac{du}{dx})}{dx} =\frac{ddu}{dx\,dx} = \frac{d^2 u}{d^2 x^2}
$$
Apparently it is not the same as the notation we were given. A $d$ is missing.
I have done some research on this and found some vague comments about "There are reasons for that, but you do not need to know…" or "That is mainly a notation issue, but you do not need to know further."
So what I am asking for is: Is this really just a notation thing?
If so, does this mean we can actually NOT use d like a variable?
If not, where does the $d$ go?
I found this related question, but it does not really answer my specific question. So I would not see it as a duplicate, but correct me if my search has not been sufficient and there indeed is a similar question out there already.
Best Answer
Physicist checking in. All the other answers seem to focus on whether $d$ is a variable and are neglecting the heart of your question.
Simply put, $dx$ is the name of one thing, so in your example
$$\frac{d^2u}{dx^2}=\frac{d^2u}{\left(dx\right)^2}$$
In your words, the "second $d$" is inside the implied parentheses.