The Language of Math: “Derivative”

The Greek letter Delta Δ can be used to denote the concept of change.

For example, when calculating the slope of a line, the letter Δ denotes the “change” or “difference” between the x and y values at two separate points:

According to Wikipedia,

Delta is the initial letter of the Greek word διαφορά diaphorá, “difference”.

I think that’s absolutely stunning. It may also seem obvious when you think about it after the fact. The change in y and the change in x are called the “difference,” and the Greek word for difference starts with the letter delta. Thus, delta was chosen to denote this difference.

The small delta δ is also frequently used in calculus to denote derivatives. Again, according to Wikipedia, calculus was developed as a way to investigate the change that frequently occurs in the natural sciences.

Wikipedia encourages us to think about it in this way:

geometry: the study of shape

algebra: the study of arithmetic operations

calculus: the study of continuous change

If we focus on just the definition of “change” and how it relates to calculus, then one can easily see how one of the fundamental branches of calculus relates to this letter.

Differential calculus explicitly concerns rates of change. To put it differently (haha, see what I did there?):

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change.

Thanks to a lovely animated math series on YouTube, I’ve been able to gain a better understanding of this concept of “rate of change.”

I’ve always understood the derivative to be the slope of a line tangential to a point on a curve. But this was purely a definition I had memorized. I’d never truly understood what this meant, or why we wanted to do this.

Through “The paradox of the derivative,” I was able to see exactly what was going on at that “point,” and learned why calling it a “point” could be confusing and misleading. Instead, what happens is that we take the slope of two points that are separated by such a minuscule distance that they may as well be called a single “point.” In theory, we are calculating the slope of this point. Yet in practice, we are calculating the slope of the line between two points—which is, again, a form of change.