75-Archetype SyllabusMathematics 2 • ARCH-MM-08 • Spec MM6

Full 6-Card Lesson 15-Question Timed Drill

UAT-UK SPEC CODE: MM6 (Differentiation)ASSUMED MATHS 1 PREREQUISITE: M4.2FREQUENCY: HIGH

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

Deconstructed from 24 past UAT-UK questions (5.85% of all historical Mathematics 2 papers). Students using standard A-Level classroom methods spend an average of 155s on this archetype—blowing past the 90s ESAT budget. Using the 2-line Sub-90 heuristic below compresses solve time to 45s (saving 110s per question).

Historical Frequency

24 Past Qs

5.85% of Mathematics 2

Classroom Trap Time

155s

+65s over 90s budget

Sub-90 Target Time

45s

Banks +110s on the clock

Dominant UI Format

5-Option A–E

Maths 1 Skill: M4.2

Card 1 & Card 2 Preview: The Classroom Time Trap vs. Sub-90 2-Line Rule

Excerpted from our interactive 6-Step Topic Lesson for ARCH-MM-08.

Launch All 6 Active-Recall Cards

CARD 1 OF 6 • THE CLASSROOM TIME TRAP (155s)

Why Students Bleed Time:

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).

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.

CARD 2 OF 6 • THE 2-LINE SUB-90 HEURISTIC (<45s)

**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).

Sanity & Limit Check: Verify units/signs against fence_post_or_domain_trap (Differentiating numerator and denominator separately: d/dx(f/g) = f'/g') and sign_or_direction_flip (Differentiating x^(-3/2) as (-3/2)x^(-1/2) instead of (-3/2)x^(-5/2)).

Examiner Distractor Blueprint for ARCH-MM-08

FOIL #1fence_post_or_domain_trap

Differentiating numerator and denominator separately: d/dx(f/g) = f'/g'

Failing to split and expand rational fractions into individual powers x^n before differentiating.

Open Fence-Post / Domain Trap Clinic

FOIL #2sign_or_direction_flip

Differentiating x^(-3/2) as (-3/2)x^(-1/2) instead of (-3/2)x^(-5/2)

Adding 1 instead of subtracting 1 from a negative fractional exponent.

Open Sign / Direction Flip Clinic

FOIL #3sign_or_direction_flip

Using tangent gradient m_t instead of normal gradient -1/m_t

Forgetting to take the negative reciprocal for the normal line.

Open Sign / Direction Flip Clinic

FOIL #4intermediate_stop_trap

Finding x-intercept of the tangent instead of the area of the triangle bounded by axes

Stopping after finding the intercept coordinates.

Open Intermediate Stop Trap Clinic

Try 1 Live Sample Question Under the 90-Second Stopwatch

Press W to open the 2-line dry-wipe scratchpad, select A–E, and lock in to reveal the Dual-View solution.

Full 6-Card Lesson Launch All 15 Questions (ARCH-MM-08)

ARCH-MM-08: Power-Rule Differentiation of Expanded Rational Powers (x^n) & Tangent/Normal Geometry (Sample)Mathematics 2

00:00.0

SUB-90 ZONE (<45S)

0.0s45s Sub-9090s ESAT Budget150s Classroom

Question 1 of 1ARCH-MM-08 (6-Card Lesson)Format: Standard 5-Option

Right-click any option to cross out • Keys A–E + Enter

Question 1 of 1 • 90s Target

Ready to Start Question 1?

The question stem, diagram, and answer options are hidden behind this pre-start gate. When you click below, the question will reveal and your 90-second split stopwatch will immediately begin ticking.

Mathematics 2•ARCH-MM-08•Standard 5-Option

or press Enter to start