My Favorite One-Liners: Part 83

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Here’s a problem from calculus:

Let f(x) = x^2 e^{3x}. Find f''(x).

We begin by finding the first derivative using the Product Rule:

f'(x) = 2x e^{3x} + 3x^2 e^{3x}.

Next, we apply the Product Rule again to find the second derivative:

f''(x) = (2 e^{3x} + 6x e^{3x}) + (6x e^{3x} + 9x^2 e^{3x}).

At this point, before simplifying to get the final answer, I’ll ask my students why the 6x e^{3x} term appears twice. After a moment, somebody will usually volunteer the answer: the first term came from differentiating x^2 first and then e^{3x} second, while the other term came from differentiating e^{3x} first and then x^2 second. Either way, we end up with the same term.

I then tell my class that there’s a technical term for this: Oops, I did it again.

While on the topic, I can’t resist also sharing this (a few years ago, this was shown on the JumboTron of Dallas Mavericks games during timeouts):

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