In this question I've derived a formula for the derivative of the sinc function. The formula was apparently correct, but I was wondering if there's a recurrence relationship between the derivative of order $k$ and, even all, the derivatives of order $0,1,...,k-1$, assuming now $f(x) = \frac{g(x)}{x}$
The attempt I did was exploiting this formula
$$ f(x) = \frac{g(x)}{x} \Rightarrow D^k f = \sum_{j=0}^k \binom{k}{j}\frac{(-1)^{k-j}}{x^{k-j+1}} D^j g(x), $$
Where $D^k = \frac{d^k}{dx^k}$. Then I've tried to substitute $$ xf(x) = g(x) \Rightarrow D^k (xf(x)) = D^{k-1}f + xD^{k}f $$
But if I substitute in the previous summation I end up with nothing.. Do you think/know any reference for the specific case of the $sinc$ function instead?