I get pretty “traditional” when it comes to introducing the concept of derivatives coming from limits. I think the reason why is that there is so much great history to talk about, and while it is an easy enough topic to grasp for some students, some students struggle with the ideas and I want to make sure that they’re solid for as many students as possible.

Here’s an example of the notes that we had after a 20 minute conversation:

These were built up very slowly and they earned each and every part as a class. The only top down parts from the teacher were the setup of the function, the motivation of what we were trying to do, and finally the name of the variables. The students came up with the idea for approximating the tangent line with the secant line (informally first, and then they remembered the language from the circles unit of geometry). They came up with the slope of the secant line, the difference quotient, and how to make the approximation of the slope better. The limits work we did leading up helped with the concept of sending h to zero while not actually having to divide by zero itself. To nail home the concept of shrinking h to get a better approximation, I used this desmos sketch:

Thoughts?