Deep Dive: Differentiation Rules

Provenance: written 2026-09-29. Primary source, read and grepped directly for this topic: research/veridian/WMA13-verified-facts.md (§1 spec table, §3 assessment/formula-booklet transcription, §4 mark-scheme conventions, §5 "4. Differentiation", §6 cross-topic patterns, §8 flagged issues, and both dated differentiation-rules.ts audit-log entries in §9). Cross-checked against what already exists, specifically to keep this document additive rather than repetitive: content/veridian/curriculum/wma13/differentiation-rules.ts (full header comment read, every heading/title block and every misconception id greped), and this same folder's codex-formula-and-method-bank.md, forge-exam-technique-guide.md and signal-frequency-and-boundaries.md (greped for every existing differentiation citation). content/veridian/question-bank/wma13.ts checked for existing marked-solution ids. One new real anchor (Jun 2026 Q3(a)–(b), a quotient-rule differentiation not yet pulled into any WMA13 differentiation document) and its arithmetic were independently re-verified by hand and with sympy before being written in below, not copied from the research file's own working.

#CODEX Deep-Dive: Differentiation Rules (WMA13, spec 4.1–4.3)

#Scope, and how this relates to what already exists

This paper's differentiation topic (spec 4.1–4.3: the five standard derivatives, the product/quotient/chain rules, and $\frac{dy}{dx}=\frac{1}{dx/dy}$) is already unusually well covered in this course — the pedagogical lesson differentiation-rules.ts derives every rule from first principles and works all four of the spec's own named example expressions; the compressed codex-formula-and-method-bank.md gives the formula skeletons; forge-exam-technique-guide.md gives the cross-cutting mark-scheme mechanics. This document does not re-derive or re-quote any of that. Its job is narrower: state the complete real model in one place as a reference (not a teaching build-up), add the small number of real, sourced facts that genuinely are not written into any of those three documents yet, and — following this course's own discipline — say plainly where the real evidence runs out rather than inventing detail to fill the gaps.

Deliberately out of scope, same boundary the lesson itself draws: what a derivative is then used for (stationary points, increasing/decreasing, tangents/normals) belongs to a different topic. Spec 4.4 (exponential growth/decay, $\frac{d}{dx}[a^x]=a^x\ln a$) is exponential-growth-and-decay.ts's territory and is not repeated here, except where a real question mixes the two (noted at the boundary, not built out).

#1. The one fact that organises the whole topic

The formula booklet (Mathematical Formulae and Statistical Tables, Issue 2, Jan 2021, p.4) prints, for differentiation: $\tan kx \to k\sec^2kx$, $\sec x\to\sec x\tan x$, $\cot x\to-\csc^2x$, $\csc x\to-\csc x\cot x$, and the quotient rule's formula itself, $\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$. Everything else this topic needs — $\sin kx\to k\cos kx$, $\cos kx\to-k\sin kx$, $e^{kx}\to ke^{kx}$, $\ln x\to\frac1x$, the sum/difference rule, and the product and chain rules themselves — is on the spec's own "will not appear in the booklet" list (spec pp.21–22) and must be memorised. Of the five spec-4.1 standard functions, four must be memorised and only one ($\tan kx$) is printed; the general marking guidance's own differentiation method-mark trigger, verified verbatim, is "Power of at least one term decreased by 1" — the mechanical signature the mark scheme actually looks for to award an M mark, independent of which specific rule produced it.

#2. Spec 4.1 — the five standard derivatives and their sums/differences

Micro-example, sums and differences (illustrative, VERIDIAN-original — not sourced to any paper, included only to show all four memorised results acting together in one line): $\frac{d}{dx}\left[4e^{2x}-3\ln5x+2\cos\frac x2\right] = 8e^{2x}-\frac3x-\sin\frac x2$. Note $\ln kx\to\frac1x$ carries no $k$ at all — the constant cancels against the chain-rule multiplier ($\frac{1}{kx}\cdot k=\frac1x$) — the one result on this list where the constant genuinely disappears rather than being memorised as part of the answer.

A genuinely additive observation, not written elsewhere in this course's WMA13 material: no verified WMA13 exam question was located anywhere in the primary research archive that tests spec 4.1 on its own — a bare sum or difference of the five standard functions, untouched by a product, quotient or composition. Every real differentiation anchor the archive contains (catalogued in §6 below) tests 4.1's results only as an ingredient inside a 4.2 or 4.3 question. This is worth knowing precisely because it means 4.1 is real, examinable content with no direct evidence of ever being examined in isolation on this paper — a standalone "differentiate $3\sin4x-e^{2x}$" question is plausible from the spec but not attested.

#3. Spec 4.2 — product, quotient (as a derived rule, not a fourth one) and chain

The three rules, stated compactly (their derivation from a small-increment/limit argument, and the proof that the quotient rule is the product rule applied to $u\cdot v^{-1}$, are both already built out in full in differentiation-rules.ts and are not repeated here):

• Product: $(uv)'=u'v+uv'$.
• Quotient: $\left(\frac uv\right)'=\frac{u'v-uv'}{v^2}$ (printed in the booklet).
• Chain: $\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)$.

The spec's own guidance names the recurring shapes this rule set is tested through — "products, quotients and composition" — and gives four illustrative expressions ($2x^4\sin x$, $\frac{e^{3x}}{x}$, $\cos x^2$, $\tan^22x$), all four of which are already worked in full in the lesson. The sub-cases below are organised the same way the real evidence actually splits, with fresh short micro-examples where a new one adds anything, and a pointer rather than a repeat where the lesson has already done the work.

Chain rule alone, nonlinear inner function. $\frac{d}{dx}\left[e^{x^2}\right]=2xe^{x^2}$ — outer function $e^u$, inner $u=x^2$, $u'=2x$. (Illustrative.)

Composition inside composition (double chain rule). $\frac{d}{dx}\left[e^{\sin2x}\right]=2\cos2x\cdot e^{\sin2x}$ — the outer exponential's own argument, $\sin2x$, needs its own chain-rule differentiation before the outer rule can even be applied. (Illustrative; the lesson's own double-chain examples are $\tan^22x$ and $\cos x^2$ — a squared-trig-of-linear-argument and a trig-of-a-power respectively — this is a third shape, exponential-of-trig, added only for variety, not because the mechanism differs.)

Product of two non-composed factors, and product where both factors are themselves compositions. Fully worked in the lesson ($2x^4\sin x$; $e^{3x}\sin2x$ for the case where both factors need their own chain rule) — not repeated.

Quotient rule — a real anchor not yet pulled into any WMA13 differentiation document. The citation-currency audit that fixed modulus-and-graph-transformations.ts (§9 of the research file, 2026-09-07 entry) independently re-verified a real Summer 2026 question, June 2026 Q3, whose parts (a)–(b) are pure spec-4.2 quotient-rule content — but that audit built only part (c) (a spec-1.4 transformation trap) into its own lesson, correctly leaving (a)–(b) out of scope there. Neither differentiation-rules.ts nor codex-formula-and-method-bank.md has since picked it up. It is genuinely additive here:

$f(x)=\dfrac{4x^2-14x+15}{x-3}$, $x<3$. (a) Find $f'(x)$ in the form $\dfrac{ax^2+bx+c}{(x-3)^2}$. [4 marks, M1 A1 dM1 A1] (b) Hence find the coordinates of the turning point of the curve. [2 marks, M1 A1]

With $u=4x^2-14x+15$ ($u'=8x-14$) and $v=x-3$ ($v'=1$): $f'(x)=\dfrac{(8x-14)(x-3)-(4x^2-14x+15)}{(x-3)^2}=\dfrac{4x^2-24x+27}{(x-3)^2}$ — independently re-verified here by hand and with sympy, matching the real mark scheme exactly. The turning point needs the numerator zero: $4x^2-24x+27=0$ gives $x=\frac32$ or $x=\frac92$; only $x=\frac32$ satisfies the question's own restriction $x<3$, and $f\left(\frac32\right)=-2$, giving $\left(\frac32,-2\right)$ — again matching the scheme exactly. This is a clean, real example of the domain restriction stated in the question doing real work: without $x<3$, the turning point would be ambiguous between two candidates.

Product where one factor is itself a composed root or reciprocal-root power of a linear inner function. This is the single most exam-discriminating shape in the whole topic, confirmed by two independent real anchors four years apart — October 2020 Q3 ($(2x+3)(4x-1)^{-1/2}$) and October 2024 Q7(a) ($x^3\sqrt{4x+7}$) — both already fully worked, with their real M1/A1/dM1/A1 structure and the "roughly a third scored the simplification mark, a quarter scored all four" statistic, in differentiation-rules.ts and in codex-formula-and-method-bank.md §4.2. Not repeated here; see either.

Reciprocal-trig extension. Micro-example (illustrative, using the booklet results directly rather than re-deriving them, which the lesson already does): $\frac{d}{dx}\left[3\sec x-\cot4x\right]=3\sec x\tan x+4\csc^24x$ (using $\cot kx\to-k\csc^2kx$ with $k=4$, so $-\cot4x\to-(-4\csc^24x)=4\csc^24x$).

#4. Spec 4.3 — $\frac{dy}{dx}=\frac{1}{dx/dy}$

Statement and derivation (the chain rule applied to $g(y(x))=x$) are fully built in the lesson and not repeated. Micro-example (illustrative): if $x=y^2-6y+2$, then $\frac{dx}{dy}=2y-6$, so $\frac{dy}{dx}=\frac{1}{2y-6}$, valid everywhere except $y=3$.

Two real traps are already documented elsewhere and only pointed to here: (i) finding and inverting $\frac{dx}{dy}$ correctly but never converting the result back into the variable the question asked for (Oct 2020 Q8(ii); lesson trap answer-left-in-the-wrong-variable); (ii) on a turning-point question, setting the reciprocated expression $\frac{1}{dx/dy}$ "equal to zero" instead of setting $\frac{dx}{dy}$ itself to zero — invalid algebra even where, coincidentally, it lands on the right value (Oct 2024, forge-exam-technique-guide.md §5.2(d)).

A synthesis point genuinely new to this course's WMA13 material: the archive's third 4.3 anchor, June 2023 Q10, combines both halves of this topic in one question — part (a) is a quotient-rule differentiation of $x$ in terms of $y$ (the "quote the rule" advice is verified verbatim against this exact question), and a later part uses the result to solve $\frac{dx}{dy}=0$ for the curve's vertical tangents. A vertical tangent on $y=f(x)$ is, structurally, exactly the same event as the Oct 2024 turning-point trap above: both are the point where $\frac{dx}{dy}=0$, i.e. where $\frac{dy}{dx}$ is undefined — one question calls it "the turning point of $x=f(y)$," the other calls it "the vertical tangent of $y=f(x)$," but the underlying instruction (work from $\frac{dx}{dy}$, never zero the reciprocal) is identical. No existing WMA13 document states this connection between the two real questions; it is worth knowing because it means the "zero $\frac{dx}{dy}$ directly" rule is attested across two series in two different framings, not one series' quirk.

#5. A boundary genuinely new to this course's WMA13 material: inverse-trig differentiation is explicitly off-spec

The October 2020 Q8(ii) examiner report — already cited elsewhere in this course for its "answer left in the wrong variable" quote — also records a third solution route some candidates attempted: rewriting the equation as $y=\arcsin(e^x)$ and differentiating arcsin directly. The report names this route explicitly as "beyond the specification so should not be encouraged." Spec 2.1 requires knowing arcsin/arccos/arctan's definitions, graphs and domains, but nowhere requires differentiating them, and this is the one place in the archive where an examiner report states that boundary outright rather than leaving it to be inferred from the spec's own silence. Worth stating plainly: a WMA13 differentiation question will not require the derivative of an inverse trig function, even when one is available as a mathematically valid alternative route to the intended answer.

#6. Exam-angle synthesis: every real anchor located for this topic

Frequency counts below reuse this folder's own aggregation in signal-frequency-and-boundaries.md (itself drawn from the 13-series research archive), cross-checked against §5 of the research file directly rather than taken on trust alone.

Spec 4.2 (product/quotient/chain): seen in 7 of 13 series checked — Oct 2020 (Q3, full detail: product+chain, root factor), Oct 2021 (Q9(a), citation only — cited for "quote the rule" advice; no further question detail transcribed in the archive), Jan 2022 (Q6(a), citation only, same caveat), Jun 2023 (Q10(a), full detail: quotient rule, and see §4 above for its 4.3 half), Jan 2024 (Q7(a), citation only, same caveat), Oct 2024 (Q7(a), full detail: product+chain, root factor, second occurrence), Jun 2026 (Q3(a)–(b), full detail: quotient rule, §3 above).

Spec 4.3 ($\frac{dy}{dx}=\frac1{dx/dy}$): seen in 4 of 13 series checked — Oct 2020 (Q8(ii), full detail: wrong-variable trap), Jan 2022 (Q10(a), full detail: implicit "show that" proof), Jun 2023 (Q10, full detail: vertical tangent via $\frac{dx}{dy}=0$), Oct 2024 (full detail: turning-point-via-reciprocal trap).

Three of the seven 4.2 citations (Oct 2021, Jan 2022 Q6a, Jan 2024) exist in the research archive only as the source of a short verbatim quote about quoting formulae before use — the underlying question's own content was not transcribed there, and inventing it here would break this document's own sourcing rule, so it is left as a named gap rather than filled in.

#7. What was not found

• No standalone spec-4.1-only question (§2).
• No question testing differentiation of arcsin, arccos or arctan — explicitly, not just by absence (§5).
• The full question text for the three citation-only 4.2 anchors (Oct 2021 Q9(a), Jan 2022 Q6(a), Jan 2024 Q7(a)) is not in the research archive and is not stated here.
• No likelihood, probability or grade-boundary claim is made anywhere above — every count is "seen in N of the M series actually checked," per this course's own standing rule.

#Sources

• research/veridian/WMA13-verified-facts.md — primary source for every citation above.
• content/veridian/curriculum/wma13/differentiation-rules.ts — checked to keep this document additive; its own worked exemplars ($2x^4\sin x$, $\frac{e^{3x}}{x}$, $\cos x^2$, $\tan^22x$, $x^3\sqrt{4x+7}$, the reciprocal-trig and $\frac{dy}{dx}=\frac1{dx/dy}$ derivations) are cited, not reproduced.
• content/veridian/technique-hub/wma13/codex-formula-and-method-bank.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md — checked for existing coverage and cross-referenced rather than repeated.
• June 2026 Q3(a)–(b) arithmetic ($f(x)=\frac{4x^2-14x+15}{x-3}$) independently re-verified by hand and with sympy for this document.