Last night I asked my AP Calculus BC students to tackle an especially challenging question. They were asked to generalize what the nth derivative would be for the function defined as f(x) = x^n/(1-x)
When n = 1 the derivative calculation was pretty straightforward and we ended up with 1/(1+-x)^2 asthe derivative. When n = 2 the work to find the second derivative was not as much fun, but it was doable. We ended up with 2/(1-x)^3 as its derivative. At this point, I wanted to predict that the derivative would be of the form n/(1-x)^n and call it a day. Unfortunately, this is not the correct answer and a quick check of the solutions manual confirmed this. This, sadly, is where my first period class ended its day. When my fourth period class came in, I was eager to start where I had left off and was prepared to wow my students with feats of Algebraic endurance by taking the third derivative of x^3/(1-x).
One of my more talented students in the class is named Connor. He told me that when he got to that point in the problem on his homework that he knew that there had to be a enter way. He changed the numerator from x^n into x^n – 1 + 1. This allowed him to write the original fraction as a sum of fractions with the first fraction transforming almost magically. I took a snapshot of his work and here it is below.
Connor recognized that the first fraction was now a geometric sum whose nth derivative would be zero. This is such a lovely example of what I hope our best students can/will do. Sure, he could have bulled his way through and seen the pattern emerge. Instead he insisted that there is a better way and he searched his mind to find a connection to something he had seen before. His classmates were taken aback, as was I. Once he rewrote the numerator, I thought of factoring and talked the students through that approach. As if that was not enough awesomeness for one class, I also had a student blow my mind with an approach to an ugly trig problem. Grading beckons me now, so I wil write again soon.