Playing a bit around with WA i found this
Namely:
$$\frac {d^n}{d^nx} \left(\frac x{f(x)}\right)^{n+1}=x\left(\frac 1{f(x)}\right)^{n+1}(2)_n$$ For $n\in\mathbb{N_0}$ and $n+1\ne x$ and $x \ne 0$ and $x\ne\frac1{f(x)}$
Where $(a)_n$ is the pochhammer symbol.
It seems very weird to me that the n'th derivative could be expressed so simply.
Is this correct, and if so how could it be proven?
Typing this instead
returns itself, unevaluated. For specific values of $n$, say $n = 3$, we get
https://www.wolframalpha.com/input/?i=D%5Bx%2Ff%5Bx%5D%2C%7Bx%2C3%7D%5D&asynchronous=false
so we know that the above syntax is performing the differentiation as it is intended. My experience with Wolfram|Alpha is that if you can enter an input in the Wolfram language, it is better to do so, because natural language input can be ambiguous.