Strange claim by WA involving nth derivative

Question

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?

If I am not wrong, with the definition of the Pochhammer symbol given in your link, if $n\geq 3$, $(2)_n=2(2-1)(2-2)...=0$. — Kelenner, Sep 02 '14 at 16:30
There's something goofy with the WA evaluation you link to. Suppose we take the case of $n=2$ explicitly; the resulting evaluation is much more complicated than the form it spits out in your link. I think what it's doing is treating $f(x)$ like a constant; compare what WA reports if you take $f(x)=1$. — Semiclassical, Sep 02 '14 at 16:41
WA apparently doesn't claim that this is the ordinary derivative, but rather the "symbolic integer derivative", whatever on earth that is... Here's a similar weird result (which, by the way, was the single hit that I got from Google when searching for that phrase in quotes), with $x^x$ instead: link. — Hans Lundmark, Sep 02 '14 at 17:17
I think the problem is related to me using a general $f(x)$ or the fact that $n$ also appeared in the function taking the derivative. — Alice Ryhl, Sep 02 '14 at 17:29
Try $ \left(\frac{1}{f(x)}\right)^{(n)} $. Check related problems. — Mhenni Benghorbal, Sep 02 '14 at 20:07

score 1 · Accepted Answer · answered Sep 02 '14 at 16:57

Typing this instead

D[x/f[x],{x,n}]

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.

Strange claim by WA involving nth derivative

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