MM6: Differential Calculus, Gradients and Optimisation
Official Cambridge & Imperial ESAT specification requirements for Differential Calculus, Gradients and Optimisation on the Mathematics 2 paper. Understand the syllabus boundaries, spot distractor traps engineered by examiners, and master the 2-step Sub-90 shortcuts that compress solve time well below the 88.8-second cutoff.
Official Syllabus Subtopics
Requirements specified in the UAT-UK framework
- 1Power rule differentiation: d/dx [xⁿ] = n·xⁿ⁻¹ for all rational n
- 2Tangents (y - y₁ = m(x - x₁)) and normal lines (m(normal) = −1/m)
- 3Stationary point location (dy/dx = 0) and classification via second derivative (d²y/dx²)
- 4Strictly increasing (dy/dx ≥ 0) and strictly decreasing (dy/dx ≤ 0) functions
- 5Constrained 2D and 3D geometric optimisation problems
Sub-90 Exam Speed Tips
How top candidates solve this spec under the 88.8s clock
The normal line has gradient -1 / f’(x). Never confuse the normal with the tangent.
In optimisation problems, express the target quantity in terms of a single variable before differentiating.
Examiners engineer distractor options using common algebraic expansions and sign slips. Always test structural ratios or invariants before calculating arithmetic values.
Tested Topics in Specification MM6
Master the individual question patterns and Sub-90 shortcuts for this specification.
Power-Rule Differentiation of Expanded Rational Powers (xⁿ) & Tangent/Normal Geometry
ESAT/TMUA restricts differentiation to rational powers xⁿ: always split fractions over a single-term denominator xᵏ*√(x) into separate powers xᵃ⁻ᵏ⁻¹ᐟ² first, then apply n*xⁿ⁻¹ term-by-term.
Stationary Point Classification, Monotonic Cubics & Constrained 2D/3D Optimisation
A cubic ax³⁺bx²⁺cx+d (a>0) is increasing for all x iff its derivative 3ax²⁺2bx+c >= 0 everywhere, which reduces to discriminant (2b)² - 4(3a)(c) <= 0 => b² <= 3ac.
Ready to Master Specification MM6 Under Exam Conditions?
Practice timed questions under the authentic Pearson VUE interface with live split stopwatch pacing, dry-wipe scratchpad, and distractor forensics.