Why second order derivative is denoted like that?

Question

As intuitive and simple as it seems the definition of derivative in liebniz notation is $\frac{dy}{dx}$. But for higher orders it gets messy ... Why is the second order denoted like $\frac{d^2y}{dx^2}$ and why not $\frac{d^2y}{d^2x}$ ...
Is it just a notation or has a deeper meaning ?
my friend says it is denoted like this because : $$\frac{d(\frac{dy}{dx})}{dx}= \dfrac{d}{dx}\Bigl(\dfrac{dy}{dx}\Bigr)= \Bigl(\dfrac{d}{dx}\Bigr)^2 y = \dfrac{d^2}{dx^2}\,y =\frac{d^2y}{dx^2} $$

Answer 1

Your friend's explanation is essentially correct, but can be simplified. The $n$th derivative of $y$ can obviously be denoted $\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)^ny$. While derivatives aren't fractions, we like to pretend they are in our notation (partly because the chain rule$$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\frac{\mathrm{d}u}{\mathrm{d}x}$$is then impossible to forget, although of course that's not a proof of it). So the natural alternative notation is $\frac{\mathrm{d}^ny}{\mathrm{d}x^n}$.