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Episodes 7, 8 and 9 all did derivatives the long way: expand, cancel, divide by h, let h go to zero. It works. It always works. And it is slow.
So here it is one final time on x³, at speed, until the screen is full of work. Then the shortcut: bring the exponent down in front, subtract one from it. Done. xⁿ becomes n·x^(n−1). That's the power rule, and that's the whole thing.
Four worked problems, escalating:
x⁵ → 5x⁴ — the plain case.
3x⁴ → 12x³ — a coefficient just rides along.
√x → rewrite as x^(1/2), then (1/2)x^(−1/2).
1/x² → rewrite as x⁻², then −2x⁻³.
The last two are the point. The rule is trivial. Recognising that a root and a fraction are powers is the actual skill, and that rewrite step is where people freeze.
Two traps: the derivative of 5 is 0, not 5x⁰ a constant doesn't change, so its rate of change is zero. And this is x raised to a number, not a number raised to x. 2ˣ is a different animal, coming later.
Then a check: the measured slope of the tangent to x³ at x = 1.5, against what 3x² predicts. Same number. Two seconds instead of thirty.
Why the rule works is the next episode.
Calculus #10 of 113. Full playlist on my channel
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2026/09/20
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