What Differentiation Is — Measuring Change at an Instant

An average erases everything inside it. So the question to ask is: what is the speed at this instant?

The number the needle is pointing at. And that number is stranger than it looks.

So what do you do?

Measure over shrinking intervals: the average over an hour, over a minute, over a second, over a thousandth of a second.

The shorter the interval, the less room there is for the speed to vary inside it, so the values converge. Something like this.

And it is not only for speed. Anywhere you ask how fast something is changing right now, the same method applies: how many new cases today, how much one more unit costs to make, how steep this hill is here.

A little further in

Cover 400 km in four hours and your average speed is 100 km/h. But if you stopped at a service area your speed was zero then, and somewhere you may have been doing 120. The average erases everything that happened inside it.

So how do you define speed at an instant? Shrink the interval: a minute instead of an hour, then a second, then a thousandth. The shorter it gets, the closer the average over it comes to the instantaneous value.

Push all the way to zero, though, and the distance is zero too: zero divided by zero. This is where the problem sat for two thousand years.

Drawing it first

In fourteenth-century Paris, Oresme drew changing quantities as pictures — time along one axis, speed along the other. Obvious now; new then. He had turned change into a shape you could look at.

With Descartes' coordinates the picture became computable. Curves became equations and equations curves. 'Speed at an instant' turned into a geometric question: the slope of a curve at a point.

From secant to tangent

Join two points on a curve and the line's slope is the average rate of change between them. Bring the points together and the line settles onto the curve. In the limit, what remains is the tangent, and its slope is the instantaneous rate.

Two men, one moment

Newton built the method in the 1660s, working on motion: to compute an orbit you must know how fast and in what direction a planet moves at each instant. He called it the method of fluxions and long declined to publish.

Leibniz arrived at the same thing independently in the 1670s — and published first. Most of the notation we use is his, because he took far more care over it.

The priority dispute outlived both men, and British mathematics clung to Newton's awkward notation for over a century out of pride, falling behind the continent. A lesson in why designing good notation matters.

What it is for

The questions differentiation answers all have one shape: how fast is this changing, right now?

Anything that varies with time or with another quantity can be differentiated.

And where the slope is zero you find a maximum or a minimum — at the top of a hill the ground is level. Every problem about making something as large or as small as possible yields to this.

The question that remainsDoes speed at an instant actually exist, or is it something we defined into being so that we could calculate?