This is a bit of a strange question, but I couldn't find an answer elsewhere and decided post my own question.
I'm currently studying differential equations and had a question regarding the Leibniz notation for higher-order DE's. I'll just give an example of what is puzzling me as it is clearer and faster.
If we want the second order derivative of an equation $f(x)$, I noticed that it is written as:
$$\frac{d^2 f(x)}{dx^2}$$
Why is the $d$ not squared in the denominator? Checking the Wikipedia article for the second derivative has shown me that:
$$ \begin{align}\frac{d^2}{dx^2}[x^n] & = \frac{d}{dx} \frac{d}{dx} [x^n] \\\ & = \frac{d}{dx}[nx^{n - 1}] \\\ & = n\frac{d}{dx}[x^{n - 1}] \\\ & = n(n - 1)x^{n - 2} \end{align}$$
In order for the first line to make sense, shouldn't the left-hand side be written as
$$\frac{d^2}{d^2 x^2}[x^n]$$
so that there are also the same number of $d$'s?
This is because $dx$ is a single symbol. Even though it looks like a $d$ and an $x$, this is not the case (at least in this context).