Introduction to Differentiation
Understand the derivative as a limit and as the gradient of a tangent; differentiate polynomials from first principles.
- 1.Using first principles, find f′(x) for f(x) = x² + 3x.
- 2.Find the gradient of the tangent to y = x³ at the point where x = 2.
2–4 marks — first-principles proofs appear in most papers.
- ·Forgetting to take the limit as h→0 in first principles
- ·Algebraic errors when expanding (x+h)ⁿ
Differential Calculus
Apply product, quotient and chain rules; differentiate trigonometric, exponential and logarithmic functions.
- 1.Differentiate y = (3x² + 1)⁵.
- 2.Find dy/dx for y = x·eˣ using the product rule.
- 3.Differentiate y = ln(sin x).
6–10 marks — the highest-weighted single outcome in the course.
- ·Forgetting the chain rule when differentiating composite functions
- ·Sign errors in the quotient rule
- ·Differentiating ln(f(x)) as 1/f(x) instead of f′(x)/f(x)
Applications of Differentiation
Find stationary points; solve optimisation problems; apply rates of change in context.
- 1.Find and classify all stationary points of y = x³ − 6x² + 9x + 2.
- 2.A rectangle has perimeter 40 cm. Find the dimensions that maximise its area.
4–8 marks — optimisation and curve sketching are exam staples.
- ·Not testing the nature of stationary points (first or second derivative test)
- ·Forgetting to check endpoints in a closed-interval optimisation
- ·Not defining variables before writing an equation in word problems
Anti-differentiation and the Indefinite Integral
Understand anti-differentiation as the reverse of differentiation; find indefinite integrals.
- 1.Find ∫(3x² + 2x − 5) dx.
- 2.Find ∫cos(2x) dx.
3–5 marks.
- ·Forgetting the constant of integration (+C)
- ·Integrating trig functions with incorrect signs (e.g., ∫sinx dx = cosx instead of −cosx)
Definite Integrals
Use the fundamental theorem of calculus; evaluate definite integrals; find areas under and between curves.
- 1.Find the area enclosed between y = x² and y = 4.
- 2.Evaluate ∫₀^π sin(x) dx.
4–8 marks — area between curves is a common multi-step problem.
- ·Not taking the absolute value when calculating area below the x-axis
- ·Subtracting integrals in the wrong order for area between curves
Differential Equations
Solve first-order differential equations of the form dy/dx = f(x) and model exponential growth/decay.
- 1.The number of bacteria N satisfies dN/dt = 0.3N. If N(0) = 500, find N after 4 hours.
3–5 marks.
- ·Forgetting to apply the initial condition to find C
- ·Misidentifying growth vs decay from the sign of the rate constant
Other Advanced topics
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