Leibniz notation for high-order derivatives

What is the reason for the positioning of the superscript $n$ in an $n$-order derivative $\frac{d^ny}{dx^n}$? Is it just a convention or does it have some mathematical meaning?

Solution 1:

Several people have already posted answers saying it's $\left(\dfrac{d}{dx}\right)^n y$, so instead of saying more about that I will mention another aspect.

Say $y$ is in meters and $x$ is in seconds; then in what units is $\dfrac{dy}{dx}$ measured? The unit is $\text{meter}/\text{second}$. The infinitely small quantities $dy$ and $dx$ are respectively in meters and seconds, and you're dividing one by the other.

So in what units is $\dfrac{d^n y}{dx^n}$ measured? The thing on the bottom is in $\text{second}^n$ (seconds to the $n$th power); the thing on top is still in meters, not meters to the $n$th power. The "$d$" is in effect unitless, or dimensionless if you like that word.

I don't think it's mere chance that has resulted in long-run survival of a notation that is "dimensionally correct". But somehow it seems unfashionable to talk about this.

Solution 2:

We can unpack what is meant, mathematically, by $\dfrac {d^ny}{dx^n}$:

$$\frac{d^n y}{dx^n} = \frac{d}{dx}\left(\dfrac d{dx}\left(\frac d{dx}\cdots\left(\frac{dy}{dx}\right)\right)\right)=\frac{\underbrace{d\cdot d \cdots \cdot d}_{n\;\text{times}}y}{\underbrace{dx \cdot dx\cdots dx}_{n\;\text{times}}}$$