Simple question on the derivative of base frame

[Trying2Learn]

TL;DR Summary

The derivative of a vector in a rotating frame (or even inertial)

I apologize: despite my verbosity, this is, I hope, a simple question.)

Consider the following relationship between a rotating reference frame and an inertial reference frame (both Bold), through a rotation matrix:
(the superscript is to designate the rotating frame e(1) and the I is for the inertial)

e⁽¹⁾=e^IR⁽¹⁾

Now suppose I wish the time derivative of the rotating frame.
(I do not know how to put the dot overhead, sorry about this)
(The dot shoud be above the first e and above the R)

^.e⁽¹⁾=e^I.R⁽¹⁾

Now, did you see how I did not take the time derivative of the base frame?

Yes, PHYSICALLY, I know it is not rotating, so I do not have to do that.

But suppose I am a stubborn person (I am) who has memorized the "product rule" in calculus -- the first times the derivative of the second + the second times the derivative of the first.

And, now I see the product of a Rotation matrix (functions -- sine, cosine) and this bold base frame: e^I

If I apply the rule, strictly, I put the dot over it and assert the result is zero.

But suppose I want to be stubborn and say: "but that base frame is not a function! I only know how to use the product rule for functions"

How do I answer myself?

The ONLY guess I can make is the following:

We began with coordinate functions: x1(1), x2(t) and x3(t)

We take time derivatives as we move a point, P(x1,x2, x3)

∂P(x1,x2, x3)/∂x1,
∂P(x1,x2, x3)/∂x2,
∂P(x1,x2, x3)/∂x3

Then, we decide to create the frame by dropping the point P, notation and looking at only the partial notation

e⁽¹⁾≡∂/∂x1,
e⁽²⁾≡∂/∂x2,
e⁽³⁾≡∂/∂x3

We do this like Ted Frankel did on page (3) of this:
http://www.math.ucsd.edu/~tfrankel/the_geometry_of_physics.pdf

And while we have a frame, its three axes "began their lives as derivatives of coordinate FUNCTIONS, so we CAN use the product rule."

Can anyone advise a better way?

For example, suppose BOTH frames are moving and we want the derivative. We have to use the product rule and get the following sequence.

e⁽²⁾=e⁽¹⁾R^(2/1)

(and please forgive me for the placement of the dot--all dots should be directly overhead)

^.e⁽²⁾= ^.e⁽¹⁾R^(2/1) + e^(1).R^(2/1)

Again, how do I know (in my ignorance and stupidity), to apply a product rule (proven for functions) to a base frame? (I have no issue, by the way, with the matrix structure formulation of the time derivative -- that is not an issue).

(And if you can tell me how to get the dot directly overhead, that would be nice, too.)

 

[jedishrfu]

For Newton derivative notation

https://tex.stackexchange.com/questions/44017/dot-notation-for-derivative-of-a-vector