Applications of Differentiation
Using first and second derivatives for tangents, normals, stationary points, curve shape, optimisation, connected rates and kinematics.
How to study A-level Mathematics
Define the objects and conditions, select a representation, execute a justified method, then validate and interpret the result.
Core concepts
Concept 1
The sign of the first derivative identifies increasing or decreasing behaviour and stationary points.
Exam cue: Find all stationary candidates and test their nature rather than assuming every one is an extremum.
Concept 2
The second derivative describes the rate of change of gradient and can help classify stationary behaviour or inflection.
Exam cue: Express the target quantity in one independent variable before optimising.
Concept 3
Optimisation and connected-rate problems require a mathematical relationship plus interpretation of feasible values.
Exam cue: Include units and signs when interpreting a related rate.
Risk pitfalls and guardrails
Classifying a stationary point solely because the first derivative is zero.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Optimising over values outside the contextual domain.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Dropping a negative rate that conveys direction or decrease.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Memory anchors
Stationary Point
A stationary point has zero first derivative.
Local Maximum
At a local maximum the derivative typically changes from positive to negative.
Point of Inflection
At a point of inflection the curve changes concavity.
Optimisation
Optimisation finds feasible extrema of a stated objective.
Connected Rates
Connected rates relate derivatives of linked quantities through a common variable.
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
Find the gradient of y=x³−4x at x=2.
For the curve y = x² + 3x, determine the equation of the tangent at the point where x = 1.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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