An inequality involving $|(1/\zeta)^{(n)}(x)|$

I was comparing various functions with $|(1/\zeta)^{(n)}(x)|$ on Desmos when $x \geq 2$ the following inequality seems to be true for all values of $n$:

$|(1/\zeta)^{(n)}(x)| \leq n!/(x-1)^n$

Is there really such an inequality or can someone proof that its true for all natural numbers $n$ and for all $x\geq 2$?If the answer is no can you give a counter example?