596C.C4.1 Rethinking the Meaning of Derivative: Real Case

Geogebra tool for derivative of real functions:

0:00 Introduction
1:15 A Grad School Story
2:38 Sophistication 1: Real Tangent Vectors
5:44 LOCATIONS are not DIRECTIONS
7:07 Limit Definition of Derivative
8:05 Slope of a Tangent Line
8:43 Reframing "Slope" as "Ratio"
9:38 Slope is a Relationship of Tangent Vectors
10:55 Visualizing the Derivative as Ratio of Tangent Vectors
14:33 Derivative is a Linear Transformation

[unpeople]

I haven't come across this way of thinking of the derivative before, but it makes perfect sense now, this was really well-explained. I watched this video without having seen any of the previous material, it just popped up in my feed, so I'll have to check out the back catalog now.

[duckymomo]

I think it would be best to teach the derivative as the best linear approximation around a point so as to not rely on the intuition of slopes