Why level of noise can be magnified twice through each numerical differentiation?
Simply put: take an independent identically distributed  Gaussian noise (one observation in blue, left) and its gradient  (in green, right). The amplitude is roughly multiplied by the norm of the gradient filter (here $h=[1\,-1]$), which is $\sqrt{2}$, which you can see from the picture. 



Their average energy  $E$ thus differ by a factor of $|h\|^2=\sqrt{2}^2 = 2$. As said by Olli, a similar question was answered in 2nd order edge detectors more susceptible to noise?. The part "can be magnified twice" is important, since this does not happen for all noises or all difference filters.

More generally, a random signal $x(t)$  passing  through a time-invariant filter with frequency response $H(f)$ produces a signal $y(t)$ with  PSD:

$$ S_y(f) = S_x(f) |H(f)|^2\,. $$

Since the Fourier domain is energy-preserving, you get that 

$$\|H\|^2\ =|h\|^2=2\,.$$

This is a reason why people have developed noise-robust differentiation, often combined with smoothing (eg Savitzky-Golay filters). See examples at 
Differentiation.

Note that in the continuous case, you should take care of non integrable processes, see for instance Variance of white Gaussian noise.