IB Maths AI HL
Further Differentiation
Paper 1 & 2
~6 min read
Product Rule
When two functions are multiplied together — like ex sin x — you can’t just differentiate each and multiply the results. The product rule gives the right answer: differentiate each function in turn while leaving the other one alone, then add. Two functions, two terms.
📘 What you need to know
- The rule: if y = uv, then dydx = udvdx + vdudx.
- Dash notation: y′ = uv′ + vu′ — a quicker way to write it.
- It’s in the formula booklet.
- When to use it: a product of two functions of x (not a function-of-a-function).
- Chain rule may be needed inside: differentiating u or v might itself need the chain rule.
- Match the notation the question uses in your final answer.
Product or composite?
The product rule is for two functions multiplied. Don’t confuse it with a composite (chain rule) — the difference is whether one function is inside the other or just sitting next to it.
product — product rule
sin x · cos x
“sin x times cos x” — two functions multiplied.
composite — chain rule
sin(cos x)
“sin of cos of x” — one function inside another.
The product rule
y = uv → dydx = udvdx + vdudx (or y′ = uv′ + vu′)
✓ given in the formula booklet
🧠 “Leave one, differentiate the other — then swap”
First term: keep u, differentiate v. Second term: keep v, differentiate u. Add them. Each function gets its turn being differentiated exactly once.
Using the product rule
🧭 Recipe — the “square” method
- Identify u and v, the two functions being multiplied.
- Differentiate each to get u′ and v′ (the chain rule may be needed here).
- Apply the formula y′ = uv′ + vu′.
- Simplify if it’s straightforward or if the question wants a particular form (often by factoring).
The “square” tip: lay u, v, u′, v′ out in a 2×2 grid. The pairs you multiply sit on opposite diagonals — u with v′, and v with u′ — which stops you pairing the wrong terms.
🤔 Why two terms, not one?
In a product, both functions are changing as x changes, and each contributes to how the product grows. Holding v steady and nudging u gives one effect (vu′); holding u steady and nudging v gives another (uv′). The total rate of change is the sum of both — which is why simply multiplying the two derivatives is wrong.
Worked examples
WE 1Differentiate y = ex sin x
A product of ex and sin x. Identify u, v and differentiate.
u = eˣ, v = sin x
u′ = eˣ, v′ = cos x
y′ = uv′ + vu′ = eˣ cos x + sin x · eˣ
y′ = eˣ(cos x + sin x)
WE 2Differentiate y = 5x2 cos 3x2
Product of 5x2 and cos 3x2 — and the chain rule is needed to differentiate the cosine.
u = 5x², v = cos 3x²
u′ = 10x, v′ = −sin 3x² × 6x = −6x sin 3x² (chain rule)
y′ = 5x²(−6x sin 3x²) + cos 3x²(10x)
y′ = −30x³ sin 3x² + 10x cos 3x²
y′ = 10x(cos 3x² − 3x² sin 3x²)
WE 3Differentiate y = x ln x
A product of x and ln x.
u = x, v = ln x
u′ = 1, v′ = 1/x
y′ = x·1x + ln x · 1 = 1 + ln x
y′ = 1 + ln x
WE 4Differentiate y = x2 e3x
Product of x2 and e3x (chain rule on the exponential).
u = x², v = e³ˣ
u′ = 2x, v′ = 3e³ˣ
y′ = x²(3e³ˣ) + e³ˣ(2x)
y′ = x e³ˣ(3x + 2)
WE 5Differentiate y = (2x + 1) sin x
Product of (2x+1) and sin x.
u = 2x + 1, v = sin x
u′ = 2, v′ = cos x
y′ = (2x + 1)cos x + sin x · 2
y′ = (2x + 1)cos x + 2 sin x
💡 Top tips
- Use the square layout — write u, v, u′, v′ in a 2×2 grid and pair the diagonals.
- Label clearly what u, v, u′, v′ are before substituting.
- Chain rule inside — if u or v is composite, differentiate it with the chain rule.
- Factor to finish — a common factor (like ex or 10x) often tidies the answer.
- Match the notation the question uses (y′ vs dydx).
- Radians for any trig.
⚠ Common mistakes
- Multiplying the two derivatives — u′v′ is not the derivative of a product.
- Confusing product with composite — sin x cos x is a product (product rule); sin(cos x) is composite (chain rule).
- Forgetting the chain rule inside — e.g. cos 3x2 differentiates to −6x sin 3x2, not −sin 3x2.
- Pairing the wrong terms — keep u with v′ and v with u′.
- Sign slips — differentiating cos brings a minus into one of the terms.
Next up — Quotient Rule. The product rule handles two functions multiplied; the quotient rule handles one function divided by another, like cos 2x3x + 2. It looks similar but has a crucial difference: a minus sign in the numerator means the order matters, so you’ll need to be careful which function goes where.
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