Power-Rule Derivative: the Slope of Any Polynomial, Term by Term, With the Tangent Drawn

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Opens on Chapter 9’s example: a sales trendline, 4x³ − 5x² + 7x + 2, with its slope at x = 2.

Polynomial one letter, numbers, + − × ÷ ^ ( ), ² ³, e, π, √, ln

Slope at

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How it works

Why xⁿ → n·xⁿ⁻¹: a square of side x grows along two edges, each x long, so its area grows at 2x; a cube grows on three faces of x², so 3x². A plain number has no slope, so it drops out. A number in front stays in front (the slope of 5x² is 5 × 2x). A sum is differentiated term by term. The rule works for negative and fractional powers too: the slope of x⁻¹ is −x⁻², and of √x = x^0.5 is 0.5x^−0.5.

From the book, Chapter 9. A 10-foot square deck grows at 20 square feet per foot of side; a 4-inch cube at 48 cubic inches per inch. The trendline’s slope at x = 2 is 35. The dropped ball is doing 80 ft/s at 2.5 s; the thrown ball rises at 16 ft/s at 1 s and tops out at 42 ft at 1.5 s. A pond of radius 3 grows at 18.85 square feet per foot: its rim. Thea’s crest climbs at 4% at the start, 2% at 50 ft, 0% at the top. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.