Talking about derivatives of exponential functions

My older son is starting the section in Stewart’s Calculus book on exponential functions. We’ve already spent a couple of days talking about inverse functions and the topic for today was finding derivatives of exponential functions.

I started by asking how he thought you’d even approach trying to find the derivative of an exponential function. It has been a while since we’ve talked about derivatives, so it took a few minutes before he came to the idea of using the definition of the derivative. Once we began to approach the problem via the definition of the derivative, we found that finding the derivative of an exponential function came down to a single limit:

Next we went to Mathematica to see if we could make any sense of this limit. Without realizing it, I had an error in the code that was causing the code to output numerical approximations. My son noticed the error and had me fix it. Unfortunately fixing that error spoiled the surprise in the answer . . . whoops 😦

Now we went back to the board to finish our computations for the derivative of an exponential function. It is pretty neat to see that the derivatives of all exponential functions are related to each other in a fairly simply way.

This was a fun discussion. The follow up discussion later was a neat problem from Stewart that asked you to show that:

for every n. That problem was a nice exercise in derivatives of exponential functions and also techniques of proof.