The derivative is a velocity¶

Level: 101 → 201 · for anyone who can multiply out (3 + h)², and who has watched a car's speedometer

One line: The velocity of a moving point at one instant is the number its average velocities settle on as the time interval shrinks; that number is the derivative, and knowing it is knowing how to take the next small step.

Two arrows on a number line¶

The talk behind Euler's identity draws e^t as an arrow on a number line labelled Position, and its derivative d/dt e^t as a second arrow starting at the tip of the first, labelled Velocity. Before e appears at all, there is the question of what the second arrow is. This page answers it with motions simple enough to compute exactly.

A point moves along a number line, and x(t) is where it is at time t. Between the times t and t + h it covers the distance x(t + h) − x(t), so its average velocity over that stretch is

(x(t + h) − x(t)) / h          distance over time

A speedometer does not show an average, though. It shows the velocity at an instant, and an instant has no length: put h = 0 in the formula and it reads 0/0. The way out is to take shorter and shorter stretches and watch where the averages go.

What the program prints¶

Verified output of derivative_as_velocity.py — regenerated by tools/run_examples.py, never hand-typed.

1. A STEADY MOTION: x(t) = 5 + 3t
   The average velocity over any stretch of time, long or short:
     from t   to t + h   distance   time     average velocity
     0        1          3          1        3
     2        5/2        3/2        1/2      3
     7        7001/1000  3/1000     1/1000   3
   Always 3. A steady motion has one velocity, and every average finds it.

2. A MOTION THAT SPEEDS UP: x(t) = t^2
   The averages over [3, 3 + h] depend on h:
     h = 1          average velocity = 7              = 6 + 1
     h = 1/10       average velocity = 61/10          = 6 + 1/10
     h = 1/100      average velocity = 601/100        = 6 + 1/100
     h = 1/1000     average velocity = 6001/1000      = 6 + 1/1000
     h = 1/1000000  average velocity = 6000001/1000000 = 6 + 1/1000000
   Algebra says why: ((3 + h)^2 - 9) / h = (6h + h^2) / h = 6 + h,
   exactly. The averages settle on 6, and no single h gives 6 itself:
   the velocity at the instant t = 3 is 6. At other instants:
     t = 0   average over [t, t + 1/1000] = 1/1000     settles on 0 = 2t
     t = 1   average over [t, t + 1/1000] = 2001/1000  settles on 2 = 2t
     t = 2   average over [t, t + 1/1000] = 4001/1000  settles on 4 = 2t
     t = 3   average over [t, t + 1/1000] = 6001/1000  settles on 6 = 2t
     t = 4   average over [t, t + 1/1000] = 8001/1000  settles on 8 = 2t
   The velocity of t^2 at time t is 2t: the derivative of t^2 is 2t.

3. THE NOTATION
   d/dt x, x'(t) and dx/dt all name the velocity of x at time t, the
   number the averages (x(t + h) - x(t)) / h settle on as h shrinks:
     d/dt (5 + 3t) = 3          d/dt t^2 = 2t
   'd' is for difference: dx is a small change in position, dt a small
   change in time, and dx/dt their ratio when both are very small.

4. STEPPING WITH A VELOCITY
   Knowing the velocity lets you step forward: over a short time dt the
   position moves by about velocity . dt. Start x = t^2 at x = 0 and
   step to t = 3 using only the velocity 2t:
         3 steps of dt = 1       x reaches 6            = 6.0000
        30 steps of dt = 1/10    x reaches 87/10        = 8.7000
       300 steps of dt = 1/100   x reaches 897/100      = 8.9700
      3000 steps of dt = 1/1000  x reaches 8997/1000    = 8.9970
   The true position is 3^2 = 9. Each step uses the velocity at its
   start, which is a little too low for a motion that speeds up, and
   the shortfall is exactly 9/n: smaller steps, smaller error.

5. IN FLOATS, THE STEP CANNOT SHRINK FOREVER
   The same average for t^2 at t = 3, in floating point:
     h        average velocity       off 6 by
     1e-1     6.100000000000012      1.0e-01
     1e-2     6.009999999999849      1.0e-02
     1e-3     6.000999999999479      1.0e-03
     1e-4     6.000100000012054      1.0e-04
     1e-5     6.000009999951316      1.0e-05
     1e-6     6.000001000927568      1.0e-06
     1e-7     6.000000087880153      8.8e-08
     1e-8     5.999999963535174      3.6e-08
     1e-9     6.000000496442226      5.0e-07
     1e-10    6.000000496442226      5.0e-07
     1e-11    6.000000496442226      5.0e-07
     1e-12    6.000533403494046      5.3e-04
     1e-13    5.986322548778844      1.4e-02
     1e-14    6.217248937900877      2.2e-01
     1e-15    5.329070518200751      6.7e-01
   The error falls with h, then rises again. (3 + h)^2 and 9 agree in
   more and more leading digits, and subtracting them leaves only the
   rounding: catastrophic cancellation. Exactly, the error is h; in
   floats the best h is near 1e-8, and no h gives more than about half
   the digits of a double.

A steady motion¶

Section 1 moves a point by x(t) = 5 + 3t: it starts at 5 and gains 3 every unit of time. Every average velocity is 3, over a whole unit of time or over a thousandth of one. A steady motion has one velocity, and there is nothing to settle.

A motion that speeds up¶

Section 2 moves the point by x(t) = t². Now the averages over [3, 3 + h] depend on h: 7 over a whole unit, 6.1 over a tenth, 6.01 over a hundredth. The program computes them as exact fractions, so the pattern is not approximate, and algebra says why:

((3 + h)² − 9) / h  =  (6h + h²) / h  =  6 + h

The average is 6 + h exactly. As h shrinks the averages settle on 6, and no stretch of time gives 6 itself, because every stretch has some length. The number the averages settle on is the velocity at the instant, and it is called the derivative. At the instant t = 3 the velocity of t² is 6; the same algebra at any t gives 2t + h, which settles on 2t, and section 2 prints it for t = 0 to 4. The derivative of t² is 2t.

"Settle on" is what a calculus course calls a limit, and it makes the phrase precise. For following the talk the picture is enough: the velocity is what the average velocities over shorter and shorter times are heading for.

The notation¶

Section 3 names it. d/dt x, x′(t) and dx/dt are three spellings of the same thing, the velocity of x at time t. The d is for difference: dx is a small change in position, dt a small change in time, and dx/dt their ratio when both are very small. So d/dt (5 + 3t) = 3 and d/dt t² = 2t. On the talk's slides, d/dt e^t sits in a box labelled Velocity, and this is all the box means.

Stepping with a velocity¶

The reason to know a velocity is that it says where the point goes next. Over a short time dt the position moves by about velocity · dt:

new position  ≈  position + velocity · dt

That is the talk's picture exactly: the velocity arrow starts at the tip of the position arrow, and a short time later the position has moved a little way along it. Section 4 uses the rule to rebuild x = t² from its velocity 2t alone, starting at 0 and stepping to t = 3. Three steps reach 6, thirty reach 8.7, three thousand reach 8.997, and the true position is 9. Each step uses the velocity at its start, which is a little too low for a motion that is speeding up, and the shortfall is exactly 9/n, so smaller steps give smaller errors.

The same stepping is how the next lesson finds e, and how Euler's identity walks a point round a circle. It has a name, Euler's method, after the same Euler.

In floats, the step cannot shrink forever¶

Exactly, the average for t² at t = 3 is off by h, so a smaller h is always better. Section 5 does the same computation in floating point and finds the opposite past a certain point: the error falls to about 4 × 10⁻⁸ at h = 10⁻⁸ and then climbs, until at h = 10⁻¹⁵ the answer is 5.33. (3 + h)² and 9 agree in more and more leading digits, and subtracting them leaves only rounding. That is catastrophic cancellation, and it means a program cannot find a derivative by making h tiny; the best it can do this way is about half the digits of a double. The lessons in this chapter measure velocities with h = 10⁻⁶, near the sweet spot, and print six decimals.

Run it yourself¶

From the root of your clone of this repository:

python3 09_Calculus/derivative_as_velocity/examples/derivative_as_velocity.py

Po polsku, w skrócie¶

Punkt porusza się po osi liczbowej, a x(t) to jego położenie w chwili t. Średnia prędkość między t a t + h to przebyta droga przez czas: (x(t + h) − x(t)) / h. Prędkość w jednej chwili, tę ze szybkościomierza, dostaje się, skracając odcinek czasu i patrząc, do czego dążą średnie. Dla x = t² w chwili 3 średnia to dokładnie 6 + h, więc dąży do 6, choć żaden odcinek czasu nie daje samego 6. Ta liczba to pochodna, zapisywana d/dt x, x′(t) albo dx/dt; pochodna t² to 2t.

Prędkość mówi, gdzie punkt będzie za chwilę: nowe położenie ≈ położenie + prędkość · dt. To dokładnie obrazek z wykładu, strzałka prędkości zaczepiona na końcu strzałki położenia, i to z niego następna lekcja buduje e^t. Pułapka jest w komputerze: w liczbach zmiennoprzecinkowych h nie może maleć bez końca, bo odejmowanie prawie równych liczb zjada cyfry, i najlepsze h to około 10⁻⁸.

Auf Deutsch: Stichwörter¶

Die Geschwindigkeit in einem Augenblick ist der Wert, auf den sich Durchschnittsgeschwindigkeiten einpendeln, wenn das Zeitintervall schrumpft: die Ableitung.

Stichwörter: Ableitung, Momentangeschwindigkeit, Durchschnittsgeschwindigkeit, Differenzenquotient, Grenzwert, Differentialquotient, numerische Ableitung, Schrittweite.