what's the ordinary derivative of the kronecker delta function? What's ordinary derivative of the kronecker delta function? I have used "ordinary" in order not to confuse the reader with the covariant derivative. I have tried the following: 
$$\delta[x-n]=\frac{1}{2\pi}\int_{0}^{2\pi}e^{i(x-n)t}dt$$ 
but that doesn't work since. $x,n \in \mathbb{Z}$, while I look for the case $x \in \mathbb{R}$

May be it is already too late, but I will answer. If I am wrong, please correct me.
Let's have a Kronecker delta via the Fourier transform getting a $Sinc$ function:
$$\delta_{k,0} = \frac{1}{a}\int_{-\frac{a}{2}}^{\frac{a}{2}} e^{-\frac{i 2 \pi  k x}{a}} \, dx = \frac{\sin (\pi  k)}{\pi  k}$$
This function looks like:
Fourier transform of "1" (Sinc) and Kronecker delta (orange dots)
Calculating the derivative we get:
$$\frac{d \delta_{k,0}}{dk} = \frac{\cos (\pi  k)}{k}-\frac{\sin (\pi  k)}{\pi  k^2} = \frac{\cos (\pi  k)}{k}$$
for $k \in \mathbb Z$
On a plot it looks like
Derivative of Sinc and Kronecker delta
