## New Ways of Thinking About Calculus

I have been teaching a long time now, so when students get me to think about a new way of doing something I am always excited. A super brief post here highlighting two solution techniques suggested to me by my AP Calculus BC students in the past week or so.

We are studying inverse functions and the relationships between their derivatives. We had settled on the fact that the function y = e^x is the function that is its own derivative. We also knew how to differentiate y = ln x based on this fact about the exponential function. I asked about the derivative of the function y = 5^x. I intended to derive a pattern for this derivative using the fact that we had derived to deal with natural log functions. Instead, one of my students suggested that we should think about the e^x function and the chain rule rather than develop a new rule. He pointed out that we know that e^x will eventually be equal to 5 so 5^x is simply a new power of e. He suggested that I write 5 = e^u and then differentiate the function y = e^(ux). I was delighted by this. Rather than add new rules to remember, simply this and rely on derived facts. I will always encourage my student any time that they can whittle down the number of rules to remember. Super excited by this.

On Thursday, I had a quiz in that class and one of the questions involved a rational function. I asked my students to verify that this function had an inverse that was also a function. I expected them to take the derivative and show that it never changed sign. I expected that because that is the way that I have thought about it and because I have taught them to approach this question through this lens. One of my students instead said that if f(a) = f(b) for some values where a and b were unequal, then that function is not one to one. He solved this equation showing that it was only true if a = b and concluded therefore that the function is one to one. Delightful! I tweeted about this and one of the responses congratulated him for relying on a definition instead of a technique. I applaud this as well and I hope to remember this well enough to present it as an alternative approach to answering this question.

Super proud of my students and I love that I get to make a big deal about the fact that I am learning from them as well as them learning from me. The big message, of course, will be that we should all be learning from each other.