M3: Calculus: Differentiation and Integration

Year 1 / ASYear 2 / A-Level All Boards (AQA, Edexcel, OCR, WJEC, CCEA) AQA

Chain, product, quotient rules, implicit/parametric differentiation, integration by parts and substitution.

Exam Board:

📋 Key Definitions and Core Concepts

Product Rule: d/dx[uv] = u(dv/dx) + v(du/dx).

Integration by Parts: ∫ u (dv/dx) dx = uv - ∫ v (du/dx) dx.

Standard derivatives: d/dx[e^{kx}] = k e^{kx}, d/dx[ln x] = 1/x, d/dx[tan kx] = k sec² kx.
Parametric: dy/dx = (dy/dt) / (dx/dt); d²y/dx² = [d/dt(dy/dx)] / (dx/dt).
Separable differential equations: ∫ (1/g(y)) dy = ∫ f(x) dx + C.

Exam-Style Question

Question:

Evaluate ∫₀¹ x e²ˣ dx using integration by parts.

Model Solution & Mark Scheme:

u = x => du = dx; dv = e²ˣ dx => v = ½e²ˣ.
[½x e²ˣ]₀¹ - ∫₀¹ ½e²ˣ dx = ½e² - [¼e²ˣ]₀¹ = ¼e² + ¼.

Q1: Differentiate y = (x² + 1) e³ˣ.

Show Model Answer

Answer: dy/dx = 2x e³ˣ + 3(x² + 1) e³ˣ = e³ˣ(3x² + 2x + 3).

Q2: Solve dy/dx = 2xy with y(0) = 3.

Show Model Answer

Answer: ∫ (1/y) dy = ∫ 2x dx => ln y = x² + C => y = 3e^{x²}.

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