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Episode 10 gave you the power rule and no reason: bring the exponent down, subtract one. Here's the reason, and it's a picture.
Take a square of side x area x². Grow the side by a small amount h. Three pieces get added: two thin strips along two edges, each x by h, and one little corner square, h by h. So the added area is 2xh + h².
Divide by h, the growth per unit of side, and you get 2x + h. Now shrink h and watch. The strips get thin but stay full length. The corner shrinks in both directions at once, so it's gone long before they are. What's left is 2x.
That disappearing corner is why the answer is 2x and not something messier.
Now one dimension up. A cube of side x, volume x³. Grow the side and you get three square slabs, one per visible face, each x² by h, plus thinner edge pieces and one tiny corner cube. Divide by h, shrink h, and everything but the three slabs vanishes. 3x². Three faces, three slabs, exponent three. In n dimensions: n slabs, each x^(n−1).
Check it on x³ at x = 2: each face is 2 × 2 = 4, three of them, so 12. And 3x² at x = 2 is 12.
One correction worth having. h² isn't dropped because it's small at h = 0.1 the strips contribute 0.60, which is also small. It's dropped because it's small relative to h, and we divided by h first. That distinction is the difference between hand-waving and calculus, and it's what episode 7's algebra was doing all along.
Calculus #11 of 113. Full playlist on my channel.
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2026/09/22
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