Question about the derivative of distance vs displacment. Displacement and Distance are not exactly the same things. I have seen everywhere on the Internet that the derivative of a distance function is it's velocity function, however to my understanding this is not true. The derivative of displacement* is velocity. Is my understanding correct, and what then is the first and second derivative of a distance function.
 A: It's more correct to say that velocity is the derivative of position. It's the instantaneous measure of how position changes with respect to time. 
The difference between displacement and distance is that distance is a scalar valued function where-as displacement is a vector, it's an arrow. Suppose for example that a particle in $\mathbb{R}^2$ has position: 

Consider finding the velocity at the black point. Let's measure displacement from the origin (displacement is independent of observer). Velocity is the change of displacement over time. The change in displacement gives the yellow lines below. You can see they are secant lines. Hence in the limit you get the usual derivative:

If instead you were to consider distance, you would be computing the length of small arcs centered at the black point. 
If you take the limit of the quotient of these arcs over time you get a scalar. What's cool is that this scalar is the length of the limit vector from the quotient of the yellow lines above.
