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.