Differentiation

Revise Differentiation for Pure Mathematics 2 P2 (AS Level) — revision notes and instant AI marking.

📖 Revision notes · preview

Cambridge International AS & A Level Mathematics 9709 — Pure Mathematics 2

Differentiation

The big idea: Add five new standard derivatives to xⁿ, combine them with the product, quotient and chain rules, and you can differentiate almost anything, including curves that are not written as y = f(x).

Summary — What This Chapter Covers

d/dx e^x = e^x, d/dx ln x = 1/x, d/dx sin x = cos x, d/dx cos x = −sin x, d/dx tan x = sec²x.
With the chain rule: d/dx e^kx = ke^kx, d/dx ln(ax + b) = a/(ax + b), d/dx sin kx = k cos kx.
Product rule: (uv)′ = u′v + uv′. Quotient rule: (u/v)′ = (u′v − uv′)/v².
Parametric: dy/dx = (dy/dt) ÷ (dx/dt).
Implicit: differentiate each term with respect to x, writing d/dx(yⁿ) = nyⁿ⁻¹ dy/dx.

1. Derivatives of e^x, ln x and Trigonometric Functions

Standard derivatives

Five new results

Angles must be in radians for the trig derivatives. Each result extends with the chain rule: differentiate the outer function and multiply by the derivative of the inside.

Standard derivatives e^x → e^x · ln x → 1/x · sin x → cos x · cos x → −sin x · tan x → sec²x
e^f(x) → f′(x)e^f(x) · ln f(x) → f′(x)/f(x)

Worked Example

Differentiate y = e^3x + ln(2x − 1) + sin 4x.

Practice Question 1

Differentiate y = tan 2x − 3 cos(x/2).

Practice Question 2

Find the gradient of y = ln(x² + 1) at x = 1.

2. Product, Quotient and Chain Rules

Combining rules

Products, quotients and composites

Identify the structure first: a product of two functions, a quotient, or a function of a function. Many questions combine them, for example x e^−2x needs the product rule and the chain rule. To find stationary points, set the derivative to zero; e^anything is never zero, so factor it out.

Rules d/dx(uv) = u dv/dx + v du/dx · d/dx(u/v) = (v du/dx − u dv/dx)/v²

Worked Example

Differentiate (a) x² ln x, (b) (2x − 4)/(3x + 2).

Worked Example

Find the exact coordinates of the stationary point of y = x e^−2x.

Practice Question

Differentiate y = x e^−x².

3. Parametric and Implicit Differentiation

Parametric and implicit

When y is not given as f(x)

Parametric: x and y are both given in terms of a parameter t. Find dx/dt and dy/dt and divide. To find a tangent, find the point and the gradient at the same value of t.

Implicit: differentiate every term with respect to x. A term in y picks up a factor dy/dx by the chain rule, and a term like xy needs the product rule. Then collect the dy/dx terms and factorise.

Methods ²

🔓 Read the full Differentiation note → You're seeing the preview · sign in to read it all

What's inside

📖 Revision notes 🎯 Learn mode ✦ AI flashcards ✓ Instant AI marking 🧊 3D explorers 🧪 Experiments & simulations 📈 Progress tracking

📄 Practise Differentiation with Pure Mathematics 2 P2 past papers Every paper with its mark scheme — answer online, marked instantly. Open →

Read the full Differentiation notes free

That's the preview — create a free account to read the rest, plus flashcards and practice questions with instant AI marking. No credit card.

Unlock the full notes free →

Differentiation