Calculus Power Rule

From the creator

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

Published 2026-09-20 · Watch on YouTube ↗
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