Z0 Additional Mathematics: Differentiation Reliability (The #1 False-Competence Chapter)

What this page is

Differentiation is where Additional Mathematics looks “learned” quickly — and then collapses later.

Why?
Because many students learn differentiation as template recognition, not as a reliable Z0 execution system.

They can do:

basic textbook forms,
but fail when:
questions mix rules,
algebra needs restructuring,
the question is disguised,
time pressure rises.

This page defines Differentiation as a Z0 pocket lattice and gives inversion tests + repair routing.

Start Here:

Definition Lock

Differentiation Reliability (Z0) is the ability to:

correctly identify what rule(s) apply,
execute differentiation steps cleanly,
simplify correctly,
handle variation and mixed rules under load.

If the student can differentiate only when the function “looks like the textbook,” differentiation is still P1.

First Principles (why differentiation collapses)

1) Differentiation is a method-selection skill

The biggest failure is not “wrong derivative.”
It is choosing the wrong rule or not seeing the structure.

2) Differentiation is coupled to Algebra reliability

Even if rules are known, weak Algebra collapses simplification and ruins answers.

3) Differentiation has layered rules

Many questions require multiple rules: chain + product, quotient + chain, etc.
Template-only students fail here.

4) Exam differentiation is disguised

Real questions often hide structure. Recognition fails; understanding survives.

Differentiation Z0 Pocket Map

Pocket A: Basic derivative rules reliability

power rule
constants
sum/difference

Pocket B: Chain rule (structure recognition)

identify inner and outer functions
apply correctly
avoid “half-chain” errors

Pocket C: Product rule (multi-line discipline)

correct setup
no missing brackets
simplify safely

Pocket D: Quotient rule (structure + algebra)

numerator/denominator discipline
bracket control
simplification reliability

Pocket E: Trig derivatives (if in scope)

correct trig derivative recall
sign discipline

Pocket F: Implicit differentiation (if in scope)

differentiate with respect to x correctly
chain inside implicit terms

Pocket G: Applications (gradient, tangents, stationary points)

connecting derivative to meaning
solving reliably after differentiation

Pocket H: Simplification under load (bridge pocket)

factorising
cancelling
rewriting for clarity

Students often think they are failing “differentiation” when they are failing Pocket H.

Differentiation Inversion Tests

Inversion Test 1: Rule-Selection Test

If the student can differentiate only when the rule is obvious, they are still in recognition.

Procedure:

give 6 functions: 2 obvious, 4 disguised
ask student to label which rule applies before differentiating

Fail: wrong rule selection or hesitation.

Inversion Test 2: Blank-Page Start Test (Differentiation)

If the student cannot set up product/quotient/chain cleanly without prompts, reliability is P1.

Fail signals:

missing brackets
wrong decomposition
wrong inner/outer

Inversion Test 3: Mixed-Rules Test

If the student collapses when two rules are needed, the pocket is not yet P2.

Example:

product + chain
quotient + chain
Fail: applies only one rule or applies rules in the wrong structure.

Inversion Test 4: Algebra Coupling Test

If the differentiation is correct but final answer is wrong, Algebra is the limiting factor.

This is the most common cause of “I know differentiation but still lose marks.”

Inversion Test 5: Mild Load Stability Test

If accuracy collapses under mild timing, differentiation is not yet exam-stable.

Below-threshold signatures

“I know the rules but I don’t know which one to use.”
“I keep missing brackets.”
“My steps look correct but answer is wrong.”
“It works in practice but fails in tests.”
“When the function is weird, I panic.”

These are classic P1 signals.

Sensors (weekly differentiation diagnostics)

Sensor 1: Rule-selection accuracy

Before differentiating, student labels rule(s). Track accuracy.

Sensor 2: Bracket discipline rate

Count missing brackets in setup lines.

Sensor 3: Inner/outer correctness (chain)

Track “outer derivative correct but inner missed” errors.

Sensor 4: Simplification error rate

If simplification errors dominate, Algebra repair must run in parallel.

Sensor 5: 48-hour retention micro-quiz

Differentiation decays fast when learned as templates. Retest after 48 hours closed-book.

Common false fixes

“Memorise more templates”

Templates fail under disguise and mixed rules.

“Spam topical worksheets”

Topical worksheets can hide method-selection weakness because all questions are same type.

“Ignore Algebra”

Differentiation cannot be stable without Algebra reliability.

Repair Protocol (P0/P1 → P2 Differentiation)

Step 1: Structure recognition drills (rule-selection first)

Before any differentiation, require:

identify rule(s)
identify inner/outer
rewrite the function in a clearer form

Step 2: Setup discipline for product/quotient

Train the setup line until it becomes automatic and bracket-safe.

Step 3: Chain rule reliability

Train decomposition:

let u = inner
write dy/du and du/dx (conceptually)
then compress back into one line once stable

Step 4: Mixed-rule ladder

Start:

chain only
Then:
product only
Then:
product + chain
Then:
quotient + chain

Step 5: Parallel Algebra repair

If simplification is the main error source:

run algebra micro-drills alongside differentiation.

Step 6: Mild load conditioning

Timed micro-sets only after rule-selection and setup are stable.

What P2 Differentiation looks like

selects correct rules reliably
setup lines are clean
can handle mild disguise
simplification is controlled
can complete within normal time

What P3 Differentiation looks like

handles mixed rules fast
recognises structure immediately
can reverse-engineer from gradient conditions
can self-check (dimension/structure checks)
stays stable under exam pacing

FAQ

Why is differentiation the #1 false-competence chapter?
Because it looks easy when practiced by type, but exams disguise and mix rules.

My child knows the rules — why still wrong?
Rule-selection and algebra simplification are usually the hidden failure points.

Should we rush into application questions?
Only after rule-selection and setup are P2. Applications collapse if the base pocket is P1.

Start Here for our Ministry of Education Series (CivOS/EducationOS Grade)

BukitTimahTutor Lattice Graph Block

Z0 Execution:
BTT.MAT.Z0.P.ALG.001
BTT.MAT.Z0.P.DIF.001
BTT.SEN.Z0.S.TTC.001
BTT.MAT.Z0.S.ERR.001

Z1 Support Loops:
BTT.PAR.Z1.P.HOM.001
BTT.TUI.Z1.P.SCF.001
BTT.SEN.Z1.S.DEP.001
BTT.SEN.Z1.S.FCG.001

Z2 Exam/Transition:
BTT.EXM.Z2.P.SEC.001
BTT.EDU.Z2.P.TRN.001
BTT.EXM.Z2.B.OLEV.001

Z3 Interfaces:
SG.EDU.Z3.B.SYL.001
SG.EDU.Z3.B.EXM.001
SG.EDU.Z3.B.PLC.001

Edges:
BTT.TUI.Z1.P.SCF.001 BindsTo BTT.MAT.Z0.P.ALG.001
BTT.MAT.Z0.P.ALG.001 BindsTo BTT.EXM.Z2.P.SEC.001
BTT.EDU.Z2.P.TRN.001 Impacts BTT.EXM.Z2.B.OLEV.001
BTT.SEN.Z1.S.DEP.001 Impacts BTT.EXM.Z2.P.SEC.001
BTT.SEN.Z0.S.TTC.001 Observes BTT.EXM.Z2.P.SEC.001