How do I prove that a = v * dv/dx?

My book says that a = dv/dt = dv/dx * dx/dt = dv/dx * v

That looks all ok but according to this equation it seems that x (displacement) is a function of t whereas x=v * t. So how can we differentiate x without differentiating v which leads to circular reasoning?

Your book has applied the chain rule to produce:

dv/dt = dv/dx * dx/dt

Now, we can get dv/dx by:

1) Differentiate

x = vt, with respect to v.

dx/dv = t

Now, if we take the inverse of this, we can obtain dv/dx

dv/dx = 1/t

This is also proven by the fact that dv/dx is the change in velocity and if you multiply it by dv/dx, which is equivalent to dividing by the change in time, as we just proved, then you obtain acceleration.