infinitesimal sketch to "show" d/dx (e^ix)
I'm not sure if this should be in Calculus or pre-Calculus, since the problem is a lead-in to a presentation of a calculus concept, and hence implicitly contain calculus concepts, but the lead-in itself will not explicitly use calculus.
This is something of a pedagogical problem. I am trying to draw a diagram with my "infinitesimal microscope" to "show", without using the standard chain rule derivation (or the series expansion of e^ix, or Euler's identity), the instantaneous change of a rotation is that rotation plus a quarter turn
(i.e., d/dx (e^ix) = i*(e^ix), without using these symbols).
To show that the instantaneous change is perpendicular to the rotation is easy, with infinitesimal secants as the difference between vectors. However, to show that the size of the change will be the same as the size of the original rotation is not as clear, since the secants are clearly smaller than my radius in my diagram.
While I am doing it, it would also be useful if it was shown to be most convenient to represent the rotation in radians.
This is supposed to be a heuristic diagram, not a formal proof.
Any suggestions? Thanks for anything.