Question

obtain the derivative and the unit tangent vector for vector function

Answer 1

The derivative of a vector valued function gives a new vector valued function that is tangent to the defined curve.  The analogue to the slope of the tangent line is the direction of the tangent line.  Since a vector contains a magnitude and a direction, the velocity vector contains more information than we need.  We can strip a vector of its magnitude by dividing by its magnitude.