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A derivative is a function, so you can differentiate it again. Nothing new: it's the same machine, run twice.
Episode 9's car, continued. s[t] = t³ − 6t² + 9t on top, velocity in the middle, acceleration at the bottom, and one playhead through all three. Each panel is the slope of the one above it. Position is where you are, velocity is how fast and which way, and acceleration is the push you feel in your back.
The moment to watch is t = 1. The car is stopped, so velocity is zero, but acceleration is −6. It's still being pushed backward. Velocity zero doesn't mean acceleration zero. A little later the car is reversing with negative acceleration and getting *faster*, because negative acceleration doesn't mean slowing down. Same signs, speeding up. Opposite signs, slowing down.
The second derivative also shapes the curve. Where s″ is negative the graph bends downward, where it's positive it cups upward, and where s″ changes sign the bend flips. That's concavity, and episode 25 goes all the way into it.
Notation: f′, f″, f‴, then f⁽⁴⁾. In Leibniz, d²y/dx². The 2s count how many times you differentiated. Nothing is squared.
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2026/10/03
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