by movement from one distinct place in three-dimensional space to a second distinct place. This motion is a distance-change (say, of magnitude s) in a certain direction. It is not just s; it is s and a direction; it is a vector. From simplest physics, all engineering students recall: Velocity of a moving body is the first derivative with respect to time of the body's moved distance. Velocity (say, v) is an infinitesimal change of s (say, ds) within an infinitesimal time period (say, dt). This quantity also has a direction; it also is a vector. The first derivative of velocity (or the second derivative of s), everybody will recall, is acceleration (say, a); and this is a vector, too.
The instantaneous values of v and a have (non-zero) magnitude if only direction is changed with time. Likewise, the derivative of s with respect to time (velocity) or the derivative of v with respect to time (acceleration) may be zero only when both the magnitude and the direction of s and v are not changing. This is not necessarily true of scalar quantities (because direction changes are either not relevant or not possible).
When vectors are added, they must be added in space, geometrically, precisely by tagging one to-scale arrowhead symbolizing one motion, s1 long in direction, angle-1, to the next arrowhead symbolizing the next motion, s2 in size in the direction of angle-2. The resultant is not s1 + s2 but the geometrically resolved distance from the origin (rest position at 1)--across three-dimensional space defined by angle-1 and angle-2--to the final position. If the instantaneous velocity or the acceleration of this simple movement must be known, the first and second derivatives of the vector sum of s1 and s2 must be capable of determination and appreciated.
Most fundamentally, and to summarize the outcome, the derivative of any vector (say, s) is the limit of the overall change (in s) as the overall change in time approaches ze...