Why does calculus use e and radians?

1. 1. Slope against height

   This is $y = 2^x$, with its tangent at the point you can drag. Below, the slope at each point is traced as a curve, beside the height, dashed. Both grow as $x$ grows; the question is how they compare.

   Question: Drag the point along $y = 2^x$. What does its slope divided by its height do?

   It stays at $0.693$. At $x = 1$ the height is $2$ and the slope $1.39$; at $x = 2$ they are $4$ and $2.77$. The slope of $2^x$ is $0.693$ times $2^x$, so the traced curve is the dashed one scaled down.

   The figure: The curve y = 2.00ˣ with its tangent at x = 2.00: height 4.00, slope 2.77. Below, the slope of y = 2.00ˣ traced up to x = 2.00, beside the height drawn dashed; slope divided by height is 0.693 everywhere.

2. 2. The base that is its own slope

   Every base does this: the slope of $a^x$ is a fixed number times its height. A bigger base climbs faster, so its number is bigger. Change the base with its slider.

   Find the base whose slope is its height: slope divided by height within $0.01$ of $1$, the traced curve lying on the dashed one.

   Near $a = 2.72$ the slope is the height at every point. That base is $e = 2.718\ldots$, and $(e^x)' = e^x$. For any other base the number is $\log a$, the natural log: $\log 2 = 0.693$, as before.

   $$\frac{d}{dx}\,a^x = a^x \log a, \qquad \frac{d}{dx}\,e^x = e^x$$$\frac{d}{dx}{a}^x$ $=a^x\log a,$$\frac{d}{dx}{e}^x$ $=e^x$

   The figure: The curve y = 2.72ˣ with its tangent at x = 1.00: height 2.72, slope 2.72. Below, the slope of y = 2.72ˣ traced up to x = 1.00, beside the height drawn dashed; slope divided by height is 1.001 everywhere.

3. 3. Not the power rule

   A common slip treats $2^x$ like $x^2$ and brings the power down: $x \cdot 2^{x-1}$. Its slopes are drawn in amber below, and its tangent at the point is drawn in amber above.

   Question: At $x = 0$ the power rule gives $0 \cdot 2^{-1} = 0$. What is the slope of $2^x$ there?

   $\log 2 \approx 0.69$: $2^x$ rises everywhere, so its slope is never $0$. The power rule is for a variable base and a fixed power, $x^n$; in $2^x$ the base is fixed and the power varies. The amber curve is even negative for every $x < 0$, where the graph plainly rises.

   The figure: The curve y = 2.00ˣ with its tangent at x = 0.00: height 1.00, slope 0.69. In amber, the power rule's tangent there, of slope 0.00. Below, the slope of y = 2.00ˣ traced up to x = 0.00, beside the height drawn dashed; slope divided by height is 0.693 everywhere. In amber, the power rule’s slopes x · aˣ⁻¹, negative for every x below 0.

4. 4. Sine’s slope

   Sine is the other family whose slope is a copy of itself. The curve is now $y = \sin x$, with $x$ in radians, and the point is about to run along it, tracing its slope below.

   Question: As the point runs along $y = \sin x$, which curve will its slope trace?

   $\cos x$: slope $1$ at $0$, $0$ at the peak $\pi/2$, $-1$ at $\pi$. It is the dashed sine moved a quarter turn left. The slope of $\cos x$ is $-\sin x$, a further quarter turn: just past $0$, $\cos x$ falls from its peak, so its slope is negative.

   $$\frac{d}{dx}\sin x = \cos x, \qquad \frac{d}{dx}\cos x = -\sin x$$$\frac{d}{dx}\sin x$ $=\cos x,$$\frac{d}{dx}\cos x$ $=-\sin x$

   The figure: The curve y = sin x, x in radians, with its tangent at x = 6.90, of slope 0.82. Below, the slope of sin x traced up to x = 6.90, beside sin x dashed.

5. 5. Why radians

   Count $x$ in degrees instead: the same wave, with $90$ where $\pi/2$ was and $180$ where $\pi$ was. Only the numbers on the $x$-axis change. In amber is what $(\sin x)' = \cos x$ would claim at $180$: a slope of $\cos 180^\circ = -1$.

   Question: With $x$ in degrees, what is the slope of $\sin x$ at $180$?

   $-\pi/180 \approx -0.0175$. The rise is the same, but one radian of run is $57.3$ degrees, so every slope is divided by $57.3$: in degrees $(\sin x)' = \tfrac{\pi}{180}\cos x$. Radians are the unit that makes the factor $1$, as $e$ is the base that does.

   The figure: The curve y = sin x, x in degrees, with its tangent at x = 180.00°, of slope −0.0175. In amber, the line of slope cos x = −1.00 that the radian rule would give there. Below, the slope of sin x traced up to x = 180.00°, beside sin x dashed. The true slope, π/180 times cos x, lies almost on the axis; in amber, cos x itself, the slope the radian rule claims.

The film, as said, 1:48

1. 0:00

   This is $y = 2^x$ with its tangent at a point. Below, its slope is traced as a curve, beside its height, dashed.

2. 0:07

   Run the point along. Slope and height both grow, but slope divided by height stays at $0.693$.

3. 0:17

   So the slope of $2^x$ is $0.693$ times $2^x$: the traced curve is the dashed one scaled down.

4. 0:25

   Every base does this. Raise the base to $3.5$ and the factor rises to $1.25$: the slope outruns the height.

5. 0:34

   In between is the base where slope and height agree everywhere, $e \approx 2.718$. The traced curve lies on the dashed one.

6. 0:44

   So $(e^x)' = e^x$. For any other base the factor is $\log a$, the natural log: $(a^x)' = a^x \log a$.

7. 0:49

   A common slip treats $2^x$ like $x^2$ and brings the power down: $x \cdot 2^{x-1}$. Its slopes are in amber.

8. 0:57

   At $x = 0$ it gives a slope of $0$, yet $2^x$ is rising there, at $\log 2 \approx 0.69$. Left of $0$ its slopes are even negative.

9. 1:05

   Sine's slope is a copy of sine. Here is $y = \sin x$, with $x$ in radians.

10. 1:10

    Run the point along: the slope traces $\cos x$, which is $1$ at $0$, $0$ at $\pi/2$ and $-1$ at $\pi$.

11. 1:22

    That is sine moved a quarter turn left. The slope of $\cos x$ is $-\sin x$, a quarter turn more.

12. 1:28

    Now count $x$ in degrees: the same wave, with $90$ where $\pi/2$ was and $180$ where $\pi$ was.

13. 1:36

    At $180$ the slope is $-\pi/180 \approx -0.0175$. The amber line, the slip's $\cos 180^\circ = -1$, is $57.3$ times too steep.

14. 1:41

    $e$ is the base, and the radian the unit, that make the factor $1$. That is why calculus uses them.