When I first witness calculus, there was a moment of magic when my teacher was explaining differentiating by 'first principles'. After he took the limiting value as change in x tends towards zero, I was like 'Woah!, mathematicians sure are smart!'.
Well, I didn't experience that same amazement when finding the first derivative of the vector function, perhaps it's because I've already met the limit. Nonetheless, I shall explain it here and hoping to 'woah!' my audiences in the process.
VECTOR DIFFERENTIAL CALCULUS >
THE FIRST DERIVATIVE
May want to check out:
VECTOR FUNCTION OF ONE VARIABLE
PARAMETER IN TERMS OF ARC LENGTH
Similar to:
VIDEO LECTURE
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CONCEPT
We all know that the first derivative is the tangent to the curve at a given point on the curve. Using a similar argument, we are incline to think that the first derivative
 is the tangent vector to a given curve defined by the vector function. To understand this, we will look at the ‘first principles’ of this derivative.
Consider, from the parallelogram law, the vector
which is represented by the arrow from the point
Since
is also along the line from
When we take the limit
This justifies our thinking that And so, when we are given the curve with the equations Lastly, in calculating All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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gives us the tangent vector to the curve where
to the point 
as shown in the diagram. 
is a nonzero scalar, the vector 
to 
. In terms of components,
, the left hand side of the above equation moves into a position tangent to the curve at 
, while the right hand side of the equation approaches 
as pictured below.
is the tangent to the curve at 
.
and ask for the tangent to that curve at the point 
, we mean the vector 
is the position vector of that curve.
, we will get a vector which, though we are not explicitly told where it starts from, should be draw at the point 
as it is after all a tangent vector. Just like how we would draw the tangent at a point of a curve in the x-y plane.
