# Is velocity the derivative of position, distance, or displacement?

All over the internet, I am seeing different people defining velocity as the derivative of either position, distance, or displacement and it is really confusing me. I can understand how the derivative of position is velocity because the very definition of velocity is (change in position)/(change in time) or (displacement)/(change in time). So how could (CHANGE in displacement)/(change in time) or (CHANGE in distance)/(change in time) give you velocity as well? Could someone tell me what is the correct way to define velocity.

If I say the position of an object is $$p(t)$$, then its displacement from any arbitrary initial point $$p_0$$ is $$p(t) - p_0$$. The derivative of that, $$\frac{d}{dt}(p(t)-p_0)$$ is exactly equal to $$\frac{dp}{dt}$$, which is the derivative of $$p(t)$$ as well.