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

Question

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 ).

Answer 1

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 $$\frac{(-1)^{n} n!}{y^{n+1}}=(-1)^n (n!) x^{n+1}.$$