Key Idea: Three rules let you differentiate anything built from simpler functions — composites, products and fractions. They're pure Paper 1 by-hand skills, and the first job in every question is to spot which rule to use.
🧭 Which rule, when?
| Shape of the function | Rule to use | What you do |
|---|---|---|
| Composite — a function inside another, f(g(x)), e.g. (x²+3)⁴, sin(3x) | Chain | Differentiate the outside, × the inside's derivative. |
| Product — two factors multiplied, u·v, e.g. x·eˣ, (2x+1)(x²−3) | Product | u′v + uv′ — both ways round, added. |
| Quotient — one thing over another, u/v, e.g. (x+1)/(x−2) | Quotient | (u′v − uv′)/v² — top derivative first, then minus. |
🔗 The three rules
- the outer function (the wrapper)
- the inner function — its derivative g'(x) is the extra factor
- the two factors being multiplied
- their derivatives — label all four before you substitute
- the top (numerator)
- the bottom — the answer is over v²
Know these: sin x → cos x, cos x → −sin x (the minus!), eˣ → eˣ (unchanged), ln x → 1/x. For composites the inner derivative is the multiplier: (ax+b)ⁿ → n(ax+b)ⁿ⁻¹ × a, sin(ax+b) → a cos(ax+b), eᵃˣ⁺ᵇ → a·eᵃˣ⁺ᵇ. If a product/quotient factor is itself composite (e.g. e²ˣ), differentiate that part with the chain rule.
✏️ IB-style worked examples
IB-style question — differentiate using the chain rule
Differentiate y = (2x² − 5)³.
Step by step:
Outer is ( )³, inner is 2x² − 5. Differentiate the outer.
Multiply by the inner's derivative (4x).
dy/dx = 12x(2x² − 5)².
IB-style question — differentiate a product
Differentiate y = x³·e²ˣ.
Step by step:
u = x³ (u′ = 3x²), v = e²ˣ (v′ = 2e²ˣ by the chain rule).
Factor out the common e²ˣ and x².
dy/dx = x²e²ˣ(3 + 2x).
IB-style question — differentiate a quotient
Differentiate y = (3x − 1)/(x² + 2).
Step by step:
u = 3x−1 (u′ = 3), v = x²+2 (v′ = 2x). Apply (u′v − uv′)/v².
Expand the numerator carefully (mind the minus).
dy/dx = (−3x² + 2x + 6)/(x² + 2)².
Important: Chain: never drop the × (inside)′ factor — the × 4x, the × 3 in (3x−1)⁵. Quotient: it's u′v − uv′, not the other way round, all over v² — getting the order wrong flips every sign.
Tap each card to reveal the answer.
Which rule for y = (x² + 1)⁵? Chain — it's a composite (something inside a power).
Differentiate y = sin(4x) 4cos(4x) — cos of it, times the inner derivative 4.
Differentiate y = (x + 2)(3x − 1) 6x + 5 — product rule: 1(3x−1) + (x+2)(3) = 6x + 5.
Differentiate y = x·ln x ln x + 1 — u′v + uv′ = 1·ln x + x·(1/x).
Differentiate y = (x − 1)/(x + 1) 2/(x + 1)² — (1(x+1) − (x−1)(1))/(x+1)² = 2/(x+1)².
What's the derivative of e⁵ˣ? 5e⁵ˣ — chain rule: the inner derivative 5 comes out front.
Exam tips
- Spot the shape first: composite → chain, product → product, fraction → quotient.
- Chain: differentiate the outside, then × the derivative of the inside (don't forget it).
- Standard derivatives: cos x → −sin x (sign!), eˣ stays eˣ, ln x → 1/x.
- Product u′v + uv′ and quotient (u′v − uv′)/v²: label u, v, u′, v′ before substituting.
- Quotient order matters — top derivative first, then minus, all over v². Factorise to tidy.