From my understanding of basic Calculus (which could very well be completely flawed), the derivative of position with respect to time would give us the slope at every point of that function, which would be the speed at that point.
In that case, there is no notion of magnitude of vector since the slope is just a measure of inclination of the tangent line at that point.
I don't understand how the magnitude of a velocity vector in a vector-valued function gives us the speed at a point, the equivalent of the slope of a tangent line in a "normal" non-vector-valued function. Why does that make sense?
Sorry if the functional-analysis tag is wrong, I wasn't sure but it seemed to fit the subject.
Best Answer
Velocity is the instantaneous speed with its direction. So the magnitude of its derivative will give you the speed which is actually a scalar quantity. But the direction of the tangent will give you the direction of the velocity.