On November 5th, 2020, thanks to the Lawrenceville Math Club, I was given an opportunity to give a math colloquium on any topic in one of their meetings. Out of all the possible topics, I decided to talk about tower functions and the derivative of infinite tower functions. Why? In one of my recent Calculus classes, I came across this problem: Find dy/dx for y=x^x (x>0). Now, those of you who take calculus will notice straight away that with implicit differentiation, this can be solved in less than three minutes. But during the last minutes of class, I wondered about how we could deal with not only finite but infinite tower functions; what if we were looking for dy/dx for y=x^x^x^x....? Or, how do we find dy/dx for y=(a^x)^(a^x)^(a^x)... where a is a constant? These questions became the starting point for my short yet intriguing exploration into tower functions and their derivatives.
Since it was my first time giving a colloquium, not all parts of the talk were perfect (time, some technological difficulties, etc). Yet, it was a great experience to share some of my thoughts with the math community and collectively work on thought-provoking and challenging tower function questions towards the latter half of the presentation. Moreover, after the end of the colloquium, I had the chance to follow up with a few people who had further questions about the topic and some of the questions we were working on; it was awesome to connect with new people on-campus that I wouldn't have met otherwise and share the same passion for math.
The calculus-based solutions to the derivatives of both finite and infinite tower functions can be found HERE or you can refer to them down below!
The solutions were written up using LaTeX, a phenomenal and easy software system for people interested in typing up their math and sharing it with others. If interested, I highly recommend checking out Overleaf, which is an online LaTeX editor you can use for free to get started.
Thanks as always for reading and remember that there's always a place in this world where you can develop and expand your passions!
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