# Getting the velocity of self-diffusion from the einstein relation

I have been reading about self-diffusion of a particle here: http://www-hsc.usc.edu/~rfarley/Diffusion-SI-2003.pdf and on page 9 it gives Einstein's relation as

$(\Delta x)^2 = 2 D t$

and I can estimate roughly how far the particle will have diffused after time $t$.

Since this is self-diffusion, the diffusion velocity should be constant, but from the above equation

$\frac{dx}{dt} = \sqrt\frac{D}{2t}$

so the particle diffuses with infinite speed at $t=0$, and the velocity decreases with time.

Clearly my understanding is wrong. Why is it not possible to differentiate the Einstein equation to get the diffusion velocity like this?