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Episode 8 said the derivative is the slope of a tangent. This says it's a rate. Same number, the word you use depends on what the axes mean.
Put time across and position up: s(t) = t³ − 6t² + 9t. Now the tangent's slope is a velocity, and a speedometer needle can be driven straight off it. The car drives forward, slows, stops, reverses all the way back to where it started, stops again, and pulls away forward. The needle crosses into negative and the car physically backs up.
That flat tangent is worth a second look. In episode 8 it was a local maximum. Here it's a car standing still. Same zero, two meanings.
Worked problem: when is the car at rest? s′(t) = 3t² − 12t + 9 — the shortcut for getting that is episode 10. Set it to zero, factor to 3(t − 1)(t − 3), and you get t = 1 and t = 3. Then the animation replays and the car stops dead at exactly those two times.
And the trap: average rate of change is the secant. The whole trip flattened into one number. Instantaneous is the tangent, right now. This car averaged one meter per second while doing nine at the start and minus three in the middle.
The units come from the axes. Position over time is velocity. Cost over units is marginal cost. People over years is a growth rate. The derivative is always how fast the vertical changes per unit of horizontal.
Calculus #9 of 113. Full playlist on my channel
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2026/09/19
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