Calculus Chapter 2 Lecture 16 BONUS

In this segment, Professor Greist introduces the concept of the infinite power tower, a mathematical function defined as \( x^{x^{x^{...}}} \), which extends infinitely. Despite the initial disbelief in its existence, the professor demonstrates that not only is it a real function, but it is also differentiable. The derivative is computed implicitly, starting with defining \( y \) as the function and then using logarithmic differentiation to arrive at \( \frac{dy}{dx} = \frac{y^2}{x(1 - y\log(x))} \). The professor emphasizes that while the function cannot be explicitly solved for \( x \), the computation of its derivative is straightforward. The function is confirmed to exist and be differentiable within a specific domain, \( x \) between \( e^{-e} \) and \( e^{e-1} \), with the function's value and derivative becoming particularly interesting around \( x = 1 \).

The second paragraph delves into the relevance of the infinite power tower, linking it to the Lambert W function, which has applications in biology and population dynamics, though these are not explored in detail within the lecture. The professor's intention in presenting the infinite power tower is to illustrate the power of mathematical analysis, emphasizing that with the right tools and mindset, one can derive properties of functions that are not visually representable or intuitively understandable. The focus is on the ability to compute the derivative of such a complex function using its implicit definition, showcasing the abstract nature of mathematical reasoning.