Computing Derivatives When You Only Have Two Points

A magnetic gradiometer is two magnetometers held exactly 0.5 metres apart. You walk across a field and the instrument reports not the absolute field strength, but the difference between what the top sensor reads and what the bottom sensor reads. If there’s a buried piece of iron, the lower sensor (closer to it) will read a stronger field. The gradient spikes.

This is finite difference approximation: estimating a derivative using discrete measurements. You can’t compute the true instantaneous rate of change at a point—you’re not doing calculus on a continuous function. You have two samples separated by a known distance. So you subtract and divide.

package main

import "fmt"

func gradient(lower, upper, spacing float64) float64 {
    return (upper - lower) / spacing
}

func main() {
    reading := []float64{48750.2, 48749.8, 48751.3, 48756.7, 48752.1}
    for i := 1; i < len(reading); i++ {
        g := gradient(reading[i-1], reading[i], 0.5)
        fmt.Printf("Gradient at step %d: %.2f nT/m\n", i, g)
    }
}

function gradient(lower, upper, spacing)
    return (upper - lower) / spacing
end

readings = {48750.2, 48749.8, 48751.3, 48756.7, 48752.1}
for i = 2, #readings do
    local g = gradient(readings[i-1], readings[i], 0.5)
    print(string.format("Gradient at step %d: %.2f nT/m", i, g))
end

The closer you space your sensors (or samples), the better the approximation—but hardware has limits. A 0.5-metre baseline is standard for portable gradiometers because it balances sensitivity to local anomalies against rejecting distant noise (the Earth’s field, power lines). The same trade-off appears everywhere: audio sample rate, GPS update frequency, temperature logging intervals.

Every derivative you’ve ever seen computed on measured data—velocity from position, acceleration from velocity, signal slopes, edge detection—uses this. Two points, a subtraction, a division. Calculus becomes arithmetic when the real world is discrete.