How to prove the derivative of position is velocity and of velocity is acceleration?

How has it been proven that the derivative of position is velocity and the derivative of velocity is acceleration? From Google searching, it seems that everyone just states it as fact without any proof behind it.

Velocity is defined as the rate of change of displacement with respect to time, and acceleration is defined as the rate of change of velocity with respect to time.

If we define change in position as $\delta x$ and a change in time as $\delta t$, then we, know that the average velocity over that time is given by $$\frac{\delta x}{\delta t}.$$ But, we do not want to know the average velocity, so we want to find the instantaneous velocity. In this case, we, by the definition of the derivative, want to take the limit as $\delta t\rightarrow0$. So, we have as velocity $$\lim_{\delta t \to 0} \frac{\delta x}{\delta t}=x'(t)$$ as desired. This isn't really a proof, per se, but it is a deeper explanation which perhaps you are seeking. A similar explanation applies to acceleration.