What a derivative is
A derivative is the rate of change of one thing as measured by the change in another. Pick two points on a curve and draw a line between them and you have an average rate over that stretch: a secant. Now slide the two points toward each other. As the distance between them approaches zero, the secant swings until it touches the curve at a single point and becomes a tangent, and the slope of that tangent is the rate of change at that instant. That instantaneous rate is the derivative.
You can take the derivative of a derivative. The first tells you how fast something is changing; the second tells you how fast that is changing. The change, and the change in the change.
Where the form came from
The idea came to me while I was thinking about the physics equations relating position, velocity and acceleration. Calculus tells us that the first derivative of position is velocity, and the second derivative of position is acceleration. Write those three as a chain of ratios and something appears:
Position is to velocity as velocity is to acceleration.
Which is the similar triangles poem arriving from a completely different direction. I had not gone looking for it. The ratio form of the derivative simply is the proportional structure, and once I saw that, the poem was already built — it only needed words.
The phantom function
Here is the difference between this form and a piece of mathematics. In calculus the function is explicit: a curve you could draw on a Cartesian coordinate system, with a slope you could calculate at any point.
The double derivative poem addresses the first and second derivatives of what I call a phantom function — a function that is never plotted and never specified. It is to be experienced with your intuition rather than described as a curve. You know that the thing is changing, and you know the change itself is changing, and the form insists those two facts stand in a fixed relation. It refuses to tell you what the function is.
The structure
Note that the middle term appears twice. In an ordinary similar triangles poem the four terms are distinct; here the first derivative is simultaneously the consequent of one ratio and the antecedent of the next. That repetition is the whole form. It is what makes the chain a chain.
And because the term is repeated, solving for it produces something an ordinary proportional poem never yields:
A worked example
Three phrases go into the three positions. Each is mapped into the source domain for one of the metaphors in the paradigm, using George Lakoff’s nomenclature:
- Liking IS position
- Caring IS velocity
- Daily rituals of love IS acceleration
Which gives the poem in its proportional form:
Liking is to caring as caring is to the daily rituals of love.
Like any equation it can be solved for any of its terms, and each solution is the same poem read from a different seat. Solved for the first term:
But I find it more interesting solved for Caring, because that is where the geometric mean surfaces:
Read what that commits to. Caring is not the sum of affection and habit, and it is not a compromise between them. It is their geometric mean — which means if either one goes to zero, caring goes to zero with it. No amount of liking survives the absence of the daily rituals, and no amount of ritual survives the absence of liking.
How to read it
Feel your way along the chain rather than calculating it. Liking is a position — a place you are. Caring is the rate at which that position is moving. The daily rituals of love are the rate at which that rate is changing: the acceleration, the thing that keeps caring from settling into a constant.
The phantom function is whatever it is that the three of them are derivatives of. The poem does not name it. That is the point of the form.
Double derivative poems
- Liking, Caring, Daily Rituals of Love 2007