Introduction to derivatives of inverse functions

We’ve started a new chapter in our calculus book -> Inverse functions.

After wondering a bit about how to approach this topic, I tried starting out in a different book, Spivak’s Calculus, which has a slightly more theoretical approach.

Now not sure how much the extra theory helped, but we did have a nice discussion about inverse functions this morning.

Tonight I wanted to give a few concrete examples and avoid the theory as much as possible. After a brief discussion, I started with the example of y = x^2 and found the derivative of the inverse function:

Next we moved on to y = e^x and y = \ln(x). He already has seen a bit of discussion about e^x and its derivatives, so I let him play with the ideas about inverse functions to see if he could find the derivative of \ln(x) on his own:

Next up were the inverse trig functions. Today I chose to focus on y = \arcsin(x) and y = \arctan(x). I started by showing him \arcsin(x) and how basic trig relations produced a pretty surprising derivative:

Next up was \arctan(x). I let him try this one on his own, but I rushed into it too quickly and forgot to talk about the domain. That led to a bit of confusion at the end, but overall I was happy that he was able to get the general idea.