Chain Rule
Recognise a composite function, differentiate the outer rule and multiply by the derivative of the inner rule.
How to study Higher Mathematics
Practise a skill without a calculator first, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and interpretation.
Core concepts
Concept 1
composite structure
Exam cue: What is the inner function?
Concept 2
outer derivative
Exam cue: What remains unchanged while the outer rule is differentiated?
Concept 3
inner derivative
Exam cue: What factor comes from differentiating the inside?
Concept 4
power composites
Concept 5
trig composites
Risk pitfalls and guardrails
Forgetting the inner derivative
Guardrail: Use correct notation and check coefficients, signs and the constant of integration.
Differentiating the inner expression twice
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Expanding a complicated power unnecessarily
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Memory anchors
Outer, keep inner
Differentiate the outside form while retaining its inner argument.
Multiply by inner prime
Complete the chain with the derivative of the inside.
Name u if needed
A temporary inner variable exposes the two layers.
Constants inside still count
Differentiate every term of the inner expression.
Do not expand by reflex
The chain rule is often shorter and safer.
Check by structure
The final derivative should contain an outer-derived factor and inner derivative.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Differentiate y=(2x+1)^5.
Differentiate y=(3x−4)^4.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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