TITLE: nth derivative cardinal sine function.
QUESTION [5 upvotes]: I need to find formula for $n$-th derivative of $\mathrm{sinc}(x)$. The following question isn't related to homework but it's a question that seems very challenging to me, and I take some interest in it.
I have  been trying to find the $n$th derivative but cant.
Help is appreciated, thanks.

REPLY [3 votes]: Related problems. You can get a formula as an integral representation. First we write $\frac{\sin(x)}{x}$ as an integral representation  

$$\frac{\sin(x)}{x} = - \int_{0}^{1} \cos(tx) dt. $$

Differentiating both sides of the above equation w.r.t. $x$ we have

$$ \left( \frac{\sin(x)}{x} \right)^{(n)} = - \int_{0}^{1} t^n \cos(tx+n\pi/2) dt. $$

Note: We used the identity 

$$(\cos(at))^{(n)}= a^n \cos(at+n\pi/2) $$

which is not hard to prove.