Who gives a shit about tangents?
Recently I’ve had a breakthrough in my understanding of a piece of mathematics that I had struggled with for years, mostly for the obscure way it’s presented usually. My previous post is about that.
It reminded me that the derivative is generally introduced in terms of tangents. And it makes no damn sense. Before being exposed to calculus, highschool students have not been trying to calculate tangents. Tangents just don’t come up. Perhaps they once did, but today’s high school math is much less geometric than it once was.
And yet, we come to the classical way of introducing the derivative as the slope of the tangent to a curve at a given point.
If you show the graph above to a student who has not yet taken calculus, and ask
them to estimate “the slope” informally, they might say it looks like 30 degrees
or so. You’ll have to explain to them that by “slope” you mean a ratio. If they
know trigonometry, you might say you’re looking for tan
of the inclination
angle. They might well ask, why not sin
or cos
?
You’ll also need to justify that in the curve shown above, the slope is
negative.
I took the graph from the Wikipedia page on the derivative , and I have to give them kudos for their explanation of the derivative. Their first paragraph:
In mathematics, the derivative shows the sensitivity of change of a function’s output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object’s velocity: this measures how quickly the position of the object changes when time advances.
I may quibble that they could have easily snuck the word ratio in there. Since they mention velocity and sensitivity, I think it’s not too much to ask for. Still, this paragraph easily motivates the definition of the derivative by appealing to things that are familiar.
The second paragraph of the wikipedia page is sadly standard:
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
I was looking at several introductions to calculus. All of them introduce the derivative via tangents, except Gilbert Strang’s Calculus, which uses the tangent as a second definition, like Wikipedia does.
I think I’ve calculated tangent lines to a graph as homework maybe once or twice; the homework was probably set as a means of selfjustification, or in halfassed fulfillment of Chekhov’s gun.
After learning calculus, the derivative becomes a core concept, and when thinking of tangents to curves, one might define them in terms of derivatives and not the other way around. Or more likely, one might not think of tangents at all, much less compute one^{1}.
Math has a problem with exposition. Come to think of it, so does computer science often. Those of us who love the subjects don’t have particular issues jumping over the hurdles.
But introducing a subject in a way that is so quickly jettisoned feels like a particularly bad miss.

In physics one uses proper tangents to trajectories, in the form of the velocity vector. In advanced calculus, for path integrals one computes the derivatives of parametrized paths. And in differential geometry and differential topology, the tangent space plays an important role.
The geometric significance of the 2D plot of a functionf: R → R
is questionable to begin with. ↩︎