# Still accelerating, or at full speed? The derivative
Source: https://www.footballdatascience.co.uk/learn/derivative
Published: 2026-09-16
> Still accelerating, at full speed, lost a yard of pace. Each is about what's happening at one moment, which is what the derivative measures, and it gives coaches something to train, measure and improve.
## The football question
A winger is one and a half seconds into a sprint. **Is he still accelerating? How quickly, right now?**
Commentators answer questions like that all the time:
- "He's still accelerating."
- "He's reached full speed."
- "He's beginning to slow down."
- "The defender is closing the gap."
All of those statements are really about how something is changing at a particular moment. Putting a number on "right now" is exactly what a **derivative** does.
## The concept
Here's our winger's speed, second by second (made-up example numbers, from [rate of change](/learn/rate-of-change)):
| Time | Speed |
|---|---|
| 0 seconds | 0.5 m/s |
| 1 second | 3.5 m/s |
| 2 seconds | 6.5 m/s |
| 3 seconds | 8.5 m/s |
Suppose we want to know what is happening at exactly **1.5 seconds**. We zoom in on that tiny moment. On a speed-versus-time graph, we can draw a line touching the curve at that point: a **tangent**. The slope of that line tells us the player's **instantaneous rate of change**, his acceleration at that exact moment.
That is essentially what a derivative tells us. It's written
$$a(t) = \frac{dv}{dt}$$
In plain football
- **dv** is a tiny change in speed; **dt** is the tiny slice of time it happened in.
- **dv/dt** is the slope of the tangent: how many metres per second faster he's getting, per second, at that instant.
- **a(t)** is his acceleration at time *t*. [Rate of change](/learn/rate-of-change) showed where it comes from: shrink the gap between two moments until the answer settles.
In our table, speed rises steadily between 1 and 2 seconds, so the tangent at 1.5 seconds is just that second's straight line: **3.0 m/s²**. Real sprints aren't made of straight lines, though. Speed climbs quickly at first and then levels off in a smooth curve, and the tangent's slope changes at every moment.
### What the commentary means in derivatives
| Commentary | The derivative |
|---|---|
| "He's still accelerating" | Acceleration is above zero: speed is still rising |
| "He's reached full speed" | Acceleration has fallen to about zero: the speed curve is flat |
| "He's beginning to slow down" | Acceleration is below zero: speed is falling |
| "The defender is closing the gap" | The distance between them is shrinking: its rate of change is negative |
## A football example
Take a smooth sprint from a standing start that levels off at a top speed of 9.5 m/s (made-up example numbers, using the kind of curve sports scientists fit to real sprint data). Draw the tangent at three moments:
Each gold line touches the speed curve at one moment; its slope is the acceleration right then. Steep early, flatter at 1.5 seconds, almost flat by 3 seconds as he nears top speed.
Do that at every moment and the derivative becomes a curve of its own: his **acceleration profile**.
| Time | Speed | Acceleration |
|---|---|---|
| 0 s | 0.0 m/s | 7.9 m/s² |
| 0.5 s | 3.2 m/s | 5.2 m/s² |
| 1 s | 5.4 m/s | 3.4 m/s² |
| 1.5 s | 6.8 m/s | 2.3 m/s² |
| 2 s | 7.7 m/s | 1.5 m/s² |
| 3 s | 8.7 m/s | 0.6 m/s² |
At 1.5 seconds he's still accelerating, at 2.3 m/s², but less than a third as hard as on his first step.
## The questions coaches ask
The acceleration profile answers the questions coaches actually care about:
- **How quickly do they become quick?** His first step is his hardest: **7.9 m/s²** from a standing start.
- **How long does it take them to reach maximum speed?** Strictly, he never quite gets there; the curve keeps edging closer. He reaches 95% of it, 9 m/s, after about **3.6 seconds**.
- **At what point does their acceleration begin to fall?** Straight away. In this sprint it halves roughly every **0.83 seconds**: 7.9, then about 4, then about 2.
- **Can they repeat that acceleration late in a match?**
- **Is their acceleration curve improving through training?**
The last two are the same question: compare the curve now with the curve before.
### A yard of pace
Say that late in a match, tired, our winger's acceleration fades a little faster: he takes slightly longer to reach the same top speed (made-up example numbers). His first step drops from 7.9 m/s² to 6.8.
The same top speed, a slightly slower climb. Tired, he reaches 7 m/s 0.27 seconds later, and after two seconds he's 0.86 m behind where he'd be fresh.
Top speed hasn't changed. But after two seconds of sprinting, the tired version of him has covered 8.9 m against 9.8 m fresh: **0.86 m behind**. A yard is 0.91 m. That's "he's lost a yard of pace", measured.
Show the mathsThe sprint curve, its derivative, and the distance run. Optional.
The speed curve used here is
$$v(t) = v_{\max}\left(1 - e^{-t/\tau}\right)$$
with top speed \(v_{\max} = 9.5\) m/s and \(\tau = 1.2\) s fresh (1.4 s tired). Its derivative is the acceleration:
$$a(t) = \frac{dv}{dt} = \frac{v_{\max}}{\tau}\, e^{-t/\tau}$$
so \(a(0) = 9.5 / 1.2 \approx 7.9\) m/s², and acceleration halves every \(\tau \ln 2 \approx 0.83\) s. Speed reaches 95% of top speed when \(e^{-t/\tau} = 0.05\), at \(t = \tau \ln 20 \approx 3.6\) s.
Going the other way, the distance run is the integral of speed:
$$x(t) = v_{\max}\left(t - \tau\left(1 - e^{-t/\tau}\right)\right)$$
giving \(x(2) \approx 9.75\) m fresh and \(8.89\) m tired.
## Why it matters
- **"Right now" is what matters in a duel.** Whether a defender can catch a winger depends on their accelerations in the next second, not their averages.
- **Top speed isn't the whole story.** Two players with the same top speed can have very different acceleration profiles, and the quicker starter wins most short races.
- **It's measurable.** GPS and tracking data record speed many times a second, so a player's acceleration curve can be tracked through a match and across a season.
- **It's trainable.** Derivatives don't just tell us where a player is. They help tell us how their performance is changing, moment by moment. And that gives coaches something they can actually train, measure and improve.
## Limitations
- **The sprint curve is a model.** Real sprints are close to it from a standing start, but players rarely start from a standstill in a match; they're often already moving.
- **Measured acceleration is noisy.** Small errors in position become big errors in acceleration, so tracking data is smoothed before derivatives are taken.
- **Straight lines aren't sprints.** Curved runs, changes of direction and decelerating to stop are all derivatives too, and all matter.
## Try it yourself
Watch a replay of a winger beating a full-back and pause it every half-second. Is the gap between them growing faster, growing slower, or shrinking? You've just estimated the derivative of the distance between them.
## Further reading
- [Limits and the definition of derivatives](https://www.3blue1brown.com/lessons/limits), 3Blue1Brown. How the tangent's slope comes from shrinking gaps, from the *Essence of calculus* series.
- [Tangent](https://en.wikipedia.org/wiki/Tangent), Wikipedia. The line that touches a curve at one point, and its slope as the derivative.
- [A spreadsheet for sprint acceleration force-velocity-power profiling](https://jbmorin.net/2017/12/13/a-spreadsheet-for-sprint-acceleration-force-velocity-power-profiling/), JB Morin. How sports scientists fit this kind of speed curve to real sprints, with a free spreadsheet.