### 1.

It was the beginning of seventh period. The tardy bell was about to ring. Reginald came into the room. What was he doing here? He’s in second period.

“Hi Reginald. What’s up?”

“Well… what do you have for me today?”

He had a smirk on his face that stayed glued there.

### 2.

Reginald is a bright student. He grasps the algebra almost before I finish speaking, sometimes completing my sentences. A few weeks ago, he proudly showed off some work he had been doing with limits. He’s hungry for more than algebra, so we work together a few times a week.

After it was clear that he had a solid understanding of limits, we started discussing derivatives. He had heard of them, but not in the way I had in mind. I taught him the definition of derivatives in terms of limits. We worked an example together. He ate it up. Then he did one on his own. And another.

I gave him an assignment. “Come back and show me what you find.”

### 3.

The next day he was back, going on about Taylor series and showing me his cosine and sine expansions.

“I’m not sure what this means,” he said, pointing to factorial terms like *2!* and *4!* in the denominators of some of the fractions. So we talked about them. And then we talked about taking the derivatives of the terms in the series one at a time. And then we were out of time.

“Take what you understand, and see if you can figure out what the derivative of *sin(x)* is.” I gave him a couple hints and jotted down some things on a piece of paper which I held out to him.

“Oh I don’t need that,” he said. He had already absorbed it all.

### 4.

When he returned, he held out a sheet of paper where he had written the series expansion for *sin(x)* and taken the derivative of each of the terms and done some factoring. At the bottom of the page was his result: the derivative of *sin(x)* is *cos(x)*.

The next time he came back with the derivative of *sin(-x*). And with very little prompting he was soon calculating the derivatives of *sin(kx)* and *sin(-kx)* and *cos(x) *and* cos(-x) *and* cos(kx)*. And I was running out of things for us to talk about. After all, I just teach algebra.

### 5.

So there we were. And when he asked, “Well… what do you have for me today?” with a smirk on his face, waiting for me to serve up the next calculus lesson, I was prepared.

We sketched some things on the board. I gave him the assignment of finding the derivative of *sin(x)* without using the Taylor series expansion that he had become so adept at manipulating. I suggested that he look it up online, figure out all the steps, and then come back and teach it to me.

“Hmmm…” he said, evidently intrigued at the notion that there might be another way.

It didn’t take long. He came back to report the results of his researches.

And now I really am running out of things to discuss. I don’t have the time. I think I’ll introduce him to hyperbolic trig functions, *sinh(x)* and *cosh(x)*. Maybe that will win me a week or so.