Differentiation
Interpret derivative graphs and solve problems about tangents, normals, optimisation and connected rates.
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Before you begin
Interpret derivative graphs and solve problems about tangents, normals, optimisation and connected rates.
Learning goals
- Relate a function to its derivative graphs
- Find and interpret a tangent to a curve
- Differentiate implicit and parametric relationships within stated scope
- Find and interpret a normal to a curve
- Classify stationary points using justified derivative tests
- Relate connected rates of change
- Model and solve an optimisation problem
Syllabus statements covered
- Include: graphical interpretation of f′(x) > 0, = 0, < 0 and f″(x) > 0, < 0; relating f′ and f graphs; simple implicit and parametric differentiation; first or second derivative tests for local extrema and stationary points of inflexion in simple cases; calculator or software location of extrema and approximate derivative at a point; tangents and normals including implicit and parametric curves; local extrema and connected rates of change. Exclude non-stationary points of inflexion and second derivatives of parametric functions.
- Include: graphical interpretation of f′(x) > 0, = 0, < 0 and f″(x) > 0, < 0; relating f′ and f graphs; simple implicit and parametric differentiation; first or second derivative tests for local extrema and stationary points of inflexion in simple cases; calculator or software location of extrema and approximate derivative at a point; tangents and normals including implicit and parametric curves; local extrema and connected rates of change. Exclude non-stationary points of inflexion and second derivatives of parametric functions.
Lessons
Work through them in order.
About lesson progress
Lesson marks record what you have studied. Checks show understanding separately. Guest marks come from this device.
- Function, gradient and local changeDistinguish function value from local rate of change and connect derivative tests to curve behaviour.
- Power, chain, product and quotient rulesChoose differentiation rules from the expression structure, retain real domains, and verify combined derivatives by another form.
- Trigonometric and exponential calculusUse radians, preserve composite domains and verify standard derivatives, antiderivatives and a recovered curve against its original conditions.
- Implicit and parametric tangentsFind branch-specific tangents and normals, handle zero coordinate rates, and verify slopes across curve representations.
- Stationary points, tangents and normalsInterpret derivative signs, classify stationary points, and distinguish local behaviour from extrema on a stated domain.
- Connected rates and optimisationConnect changing dimensions and net rates, then justify constrained optima using the original model and feasible domain.