After asking students to find the first, second, third, fourth, tenth, hundredth, and the five thousand eighty first derivatives of sin(x); I challenged them to write out the general derivative of sin(x). They had some problems with notation, as this is the first time they’ve been asked to generalize such work at this level, but some creative solutions (which were correct!) were put forth. Here’s a programming student in the middle of his thought process. It’s not exactly “correct”, but some fantasticÂ thought going on here:

And a bonus “problem” that some students had when finding the derivative of g(x)=x^3 * ln(pi). Is this really a problem? NO! I’d argue that this student will take to the implicit differentiation that we’ll be doing today much easier than the average student who has memorized the idea that “constants stay out in front when finding the derivative”.

How did I address this problem? It was fun, I told the student an x value, and then asked about the value of pi.

What is pi when x = 3? What is pi when x = 10? What is pi when x = 500? Oh so no matter how much I change x, pi isn’t changing?