Technique Masterclasses

Pure Mathematics 1

Deep Dive: Differentiating Negative and Fractional Indices

CODEX Deep-Dive: Differentiating Negative And Fractional Indices (WMA11, spec 4.2)

Provenance (2026-09-29). Built from a direct read of research/veridian/WMA11-verified-facts.md (§1 spec table of contents; §3 mark-scheme conventions; §4.4 Differentiation; the dated 2026-09-14 differentiating-negative-and-fractional-indices.ts mark-scheme-bullet coverage audit, re-read in full, including the "one real finding NOT added, on explicit instruction" note) — grepped specifically for "indices", "negative", "fractional", "x^(1/2)", "3√x" and "x > 0"; the full text of the existing lesson content/veridian/curriculum/wma11/differentiating-negative-and-fractional-indices.ts; content/veridian/technique-hub/wma11/codex-formula-and-method-bank.md §1.1.1 and §1.4.1–1.4.2 (checked for overlap before writing a line of this document); content/veridian/technique-hub/wma11/signal-topic-brief-differentiating-negative-and-fractional-indices.md (the existing frequency/context synthesis for this exact topic); content/veridian/technique-hub/wma11/codex-deep-dive-surds-manipulation-and-exact-answers.md and content/veridian/curriculum/wma11/surds-manipulation-and-exact-answers.ts (both already carry the real Jan 2023 Q1(b) surd-intermediate-working finding — checked so it is cited, not redeveloped, here); content/veridian/technique-hub/wma11/forge-exam-technique-guide.md and forge-error-pattern-tracker.md (checked for overlap with the trap taxonomy); content/veridian/curriculum/wma11/hidden-higher-degree-equations-from-differentiation.ts and content/veridian/curriculum/wma11/cubic-graph-sketching-from-factored-form.ts (checked for the adjacent product-expansion and domain-restriction material, so it is not redeveloped here); and content/veridian/specCoverage.ts (verbatim spec-guidance text and worked-example expressions for WMA11 spec 4.2). content/veridian/question-bank/wma11.ts was checked and contains no entry tagged to spec 4.2. No new primary-source PDF was opened for this pass — every claim below traces to material the facts bank already verified, to specCoverage.ts, or to direct algebra that is independently checkable. Not affiliated with or endorsed by Pearson Edexcel.

0. What this document adds, and what it deliberately does not repeat

This exact topic already has unusually thorough, mark-scheme-verified coverage elsewhere in this course. Fully developed, and not repeated below:

What follows is additive in three specific ways instead:

  1. A real, spec-mandated sub-case this topic's own lesson never touches — differentiating a multi-term fraction over a single power-of-x denominator, which spec 4.2's own worked-example guidance names directly, and which produces a wider spread of fractional and negative-fractional indices in one expression than any example currently in the lesson.
  2. A mathematical account, not stated anywhere else in this technique hub, of why the real anchors state "x > 0" at all — grounded in the domain of the original function itself, distinct from the already-covered trap of a derived equation's rejected root.
  3. A compact statement of where the real marks actually sit along the rewrite → differentiate → evaluate pipeline across the two confirmed anchors — a framing the SIGNAL brief's own tariff discussion does not build.

1. The rule, and the shapes spec 4.2 actually names

Spec 4.2's item is short: differentiation of xⁿ, and related sums, differences and constant multiples. It rests on spec 1.1's guarantee that a^(m/n) and (ⁿ√a)^m are the same object for all rational exponents — the equivalence that lets √x become x^(1/2) and 1/x² become x⁻² before the power rule is ever applied. The codex formula bank's own verified condition on that equivalence is worth restating here because it is load-bearing for §3 below: for a^(m/n) to be real-valued when n is even, a ≥ 0 is required.

The rule itself — f(x) = xⁿ ⟹ f′(x) = nxⁿ⁻¹ — holds without qualification for every rational n. Why it holds for n = 0, 1, 3, −1 and ½ specifically, worked from the limit definition with three genuinely different pieces of algebra (expand-and-cancel, combine-a-fraction-and-cancel, rationalise-and-cancel), is already derived in full in codex-deep-dive-the-derivative-as-a-limit.md §2 and is not re-derived here.

What matters for this document is a fact easy to miss because the existing lesson's own examples don't surface it: spec 4.2's guidance names not one but two worked-example shapes a candidate must be able to handle, and only one of them is a single term needing a rewrite. The two named expressions are a product — (2x + 5)(x − 1) — and a fraction with a multi-term numerator over a single power-of-x denominator — (x² + 5x − 3)/(3√x). Neither is the "rewrite one term, then differentiate" pattern the lesson drills.

2. The three real shapes, and which one is genuinely new here

2.1 A single root or reciprocal term — already fully covered

Rewrite the one term as a power of x, apply nxⁿ⁻¹, track the sign. This is the whole of the existing lesson's worked chain, chain-drill and marked solution, checked against both real anchors (Jan 2023 Q1, Jan 2025 Q5(a)). Nothing to add.

2.2 A product needing expansion first — adjacent, not developed here

Spec 4.1 explicitly excludes the chain rule at P1, and there is no product rule on this paper either — so a term like (2x + 5)(x − 1) must be expanded into a polynomial before the power rule can touch it term-by-term. The expanded polynomial here is pure positive-integer powers; this shape does not itself produce a negative or fractional index, so it sits just outside this topic's own title even though it shares spec 4.2. The real confirmed exam anchor for exactly this expand-then-differentiate move — f(x) = 2(x+1)(x−3)², expanded then differentiated to a stationary-point tangent problem, Oct 2021 Q6 — already belongs to cubic-graph-sketching-from-factored-form.ts and hidden-higher-degree-equations-from-differentiation.ts. Not redeveloped here.

2.3 A multi-term fraction over a single power-of-x denominator — genuinely new

This is spec 4.2's other named shape, and it is the one that actually belongs under this topic's title: dividing a multi-term numerator by a single surd or power-of-x denominator produces several terms whose indices are each shifted by the same amount, scattering positive, fractional and negative-fractional indices across one expression before any differentiation happens. No lesson in this course currently works this shape for differentiation (its mirror image for integration — a multi-term numerator over a single-term denominator, split before integrating — is fully covered by the real Jan 2023 Q3 anchor in splitting-fractions-before-integrating.ts; a student who has drilled that lesson's split step already owns most of the technique needed here, run forward instead of backward).

Method. Rewrite the denominator as a power of x, then apply xᵃ/xᵇ = xᵃ⁻ᵇ to each numerator term separately — never to the fraction as a whole — so that a numerator of degree 2, 1 and 0 becomes three terms whose indices are each the original power minus the denominator's own index.

Worked micro-example, using spec 4.2's own named expression:

(x² + 5x − 3)/(3√x), x > 0

Rewrite the denominator: 3√x = 3x^(1/2). Split and simplify each term:

(1/3)·x²/x^(1/2) + (5/3)·x/x^(1/2) − (3/3)/x^(1/2)
= (1/3)x^(3/2) + (5/3)x^(1/2) − x^(-1/2)

Differentiate term by term with nxⁿ⁻¹:

d/dx[(1/3)x^(3/2)] = (1/3)(3/2)x^(1/2)   = (1/2)x^(1/2)
d/dx[(5/3)x^(1/2)]  = (5/3)(1/2)x^(-1/2)  = (5/6)x^(-1/2)
d/dx[-x^(-1/2)]     = -(-1/2)x^(-3/2)     = (1/2)x^(-3/2)

dy/dx = (1/2)x^(1/2) + (5/6)x^(-1/2) + (1/2)x^(-3/2)
      = (1/2)√x + 5/(6√x) + 1/(2x√x)

Check, at x = 1 (chosen because every root is trivial there, so the check itself can't hide an arithmetic slip): the split form gives (1/3)+(5/3)−1 = 1, matching the original (1+5−3)/(3·1) = 1 ✓. The derivative gives (1/2)+(5/6)+(1/2) = 11/6. Cross-checked independently via the quotient rule (u = x²+5x−3, v = 3x^(1/2); u(1)=3, u′(1)=7, v(1)=3, v′(1)=1.5): (u′v − uv′)/v² = (21 − 4.5)/9 = 16.5/9 = 11/6 ✓ — both routes agree exactly.

This shape reaches an index of −3/2, one step further from zero than any term in the existing lesson (whose most extreme index is −1/2 or −2, never both a negative and a fraction stacked together this far). It is real, spec-named content — but it must be stated honestly: none of the five WMA11 series the facts bank has fully mined (Jan 2019, Oct 2020, Oct 2021, Jan 2023, Jan 2025) tests this multi-term-numerator shape under this topic. Both confirmed real anchors instead use a single-term fraction (1/x or 2/x, not a numerator needing a split first). This sub-case is required by the spec's own worked-example guidance, not yet observed on the checked papers — seen in 0 of 5.

3. Why the real anchors state "x > 0" — the domain of the function itself

Both confirmed real anchors open by stating x > 0: Jan 2023 Q1's y = 2 + 10x^(1/2) − 2x^(3/2), and Jan 2025 Q5(a)'s y = 4x³ + 2/x + 9. This is not decoration, and it is not (at this stage of the question) about which root of a later equation survives — it is the domain the original function already has, before any differentiation happens.

A term x^(1/2) is real-valued only for x ≥ 0 — the even-root condition already established in §1 above. A term 2/x = 2x⁻¹ is undefined at x = 0. Put the two conditions together on a function that mixes an even-root fractional index with a negative index, and x > 0 is the tightest single statement that keeps every term meaningful: it satisfies the fractional term's own requirement (x ≥ 0) and strictly excludes the one point the negative-index term cannot tolerate (x = 0), in one line. This is why a WMA11 function combining these two index types will essentially always need a domain statement, and why "x > 0" specifically — rather than "x ≠ 0" or "x ≥ 0" alone — is the natural choice whenever the fractional term's own root is even.

Worth being precise about scope: this same stated condition later does a second, different job in the rest of Jan 2025's own Q5 — ruling out the negative root of a derived quadratic-in-x² equation at part (c), a confirmed real trap ("despite 'x > 0' in the wording of the question") already fully covered by hidden-higher-degree-equations-from-differentiation.ts and its trap taxonomy. This section is only about the first job — why the condition is there at all, on the function as originally written — which is not stated anywhere else in this technique hub.

4. Where the real marks actually sit

The two confirmed anchors test different amounts of the same pipeline (rewrite → differentiate → evaluate exactly), and it is worth being precise about which:

So of the two confirmed real anchors, only one (Jan 2023 Q1) actually completes the rewrite → differentiate → evaluate pipeline this topic's own lede describes; the other tests just its first half as a small opening gate into a differently-themed question. Stated the way this course's own evidentiary convention requires: 1 of 2 confirmed real anchors ends in a numeric exact gradient — not both, and not a rate that should be assumed to generalise beyond these two instances.

5. Sources