Differentiation
Interpret derivatives as gradients and rates and use first and second derivatives for tangents, normals and curve behaviour.
How to prepare for the current TMUA
Secure the shared mathematical specification, practise application in unfamiliar settings, then make every Paper 2 implication and proof step explicit.
Core concepts
Concept 1
Differentiate rational powers and related sums after simplifying.
Exam cue: Rewrite roots and reciprocals as powers before differentiating.
Concept 2
Use derivative values for rates, tangent and normal gradients.
Exam cue: Separate the x-coordinate condition from the requested y-value or gradient.
Concept 3
Locate and classify maxima and minima and determine increasing or decreasing intervals.
Exam cue: Use a sign change or second derivative to classify a stationary point.
Risk pitfalls and guardrails
Treating every stationary point as an extremum without a test.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Confusing the normal gradient with the negative tangent gradient.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Using differentiation from first principles as required specification content.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Memory anchors
Power derivative
d(xⁿ)/dx = nxⁿ⁻¹.
Rate meaning
A derivative measures instantaneous change in the output per input.
Stationary
A stationary point satisfies f′(x) = 0.
Maximum sign
The gradient changes from positive to negative.
Minimum sign
The gradient changes from negative to positive.
Normal gradient
Use the negative reciprocal of a non-zero tangent gradient.
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
For f(x)=x³−6x²+9x, what is f′(4)?
The tangent to y=x²+3x at x=2 meets the y-axis at b. What is b?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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