ARCH-MM-08: Power-Rule Differentiation of Expanded Rational Powers (x^n) & Tangent/Normal Geometry

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-08: Why the 155s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~155s (Blows 90s Clock)

Rewriting

\frac{2 x^{2} + 4 x - 6}{\sqrt{x}} = 2x^{3/2} + 4x^{1/2} - 6x^{-1/2}

and differentiating term by term gives

\frac{dy}{dx} = \frac{6}{2}x^{1/2} + \frac{4}{2}x^{-1/2} - \frac{-6}{2}x^{-3/2}

. Substituting

x = 1

yields

8

(~125s).

The Fast ShortcutTarget: 45s (Saves 110s)

**Line 1 (Split Over

\sqrt{x}

):** Write

y = 2x^{3/2} + 4x^{1/2} - 6x^{-1/2} \implies y' = \frac{6}{2}x^{1/2} + \frac{4}{2}x^{-1/2} - \frac{-6}{2}x^{-3/2}

.
**Line 2 (Substitute

\sqrt{1} = 1

):**

y'(1) = 8

(30s).

Where Students Lose Time: In 24 past UAT-UK questions (5.85% of Mathematics 2), candidates lose 110s by failing to split and expand rational fractions into individual powers x^n before differentiating. Instead, apply MM6 with Maths 1 skill M4.2.

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