Q:

n-th derivative with respect to $\frac{1}{x}$

Is it any easy way to calculate : 
$\frac{d^n x}{d\left(\frac{1}{x}\right)^n}$ 
for arbitrary $n\in\mathbb{N}$ ?
(for $n=1$ it is obvious, but for $n>1$ the formula for $n$-th derivative of composition is more complicated ).  

A:

If $y=1/x$, then you ask about $\frac{d^n x}{dy^n}$, probably rewritten as a function of $x$. Clearly, $x=1/y$ and the $n$th derivative is then $$\frac{(-1)^{n} n!}{y^{n+1}}=(-1)^n (n!) x^{n+1}.$$

