The Standard Deviation of The Derivative of a Signal Given a signal with zero mean and a standard deviation of 0.1 sampled at 5000 Hz.
What would be the Standard Deviation of its 1st, 2nd and 'n' derivative?

For instance, let's say we measure the distance to a constant range target.
The STD of the sensor measurements of the range is 0.1 [Meter] (Assume the error is white) and its sampling rate is 5000 Hz.

What would be the standard deviation of the acceleration process?

Thanks.
As pointed out in pichenettes' answer, you need to know more about the original signal $x(t)$. If the signal's power spectrum $S_x(\omega)$ is known then the power spectrum of its $n^{th}$ derivative is

$$S_y(\omega)=\omega^{2n}S_x(\omega)$$

The power of the $n^{th}$ derivative can then be computed as

$$E\{|y(t)|^2\}=\frac{1}{2\pi}\int_{-\infty}^{\infty}\omega^{2n}S_x(\omega)d\omega\tag{1}$$

(assuming convergence of the integral, e.g. by assuming that $x(t)$ is band-limited). Since the mean of the derivative is zero (even if the mean of the original signal isn't), the variance of $y(t)$ equals its power, and so the standard deviation is the square root of the expression in (1).