IB Maths AI HL
Further Differentiation
Paper 1 & 2
~6 min read
Quotient Rule
When one function is divided by another — like cos 2x3x + 2 — you use the quotient rule. It looks like the product rule’s cousin, but with one crucial twist: a minus sign in the numerator, which means the order of the two functions matters. Get the order wrong and the sign flips.
📘 What you need to know
- The rule: if y = uv, then dydx = vdudx − udvdxv2.
- Dash notation: y′ = vu′ − uv′v2.
- It’s in the formula booklet.
- Order matters: because of the minus, vu′ comes first — u is the numerator, v the denominator.
- When to use it: a fraction where both top and bottom are functions of x.
- Chain rule may be needed inside to differentiate u or v.
When to reach for it
The quotient rule is for a fraction where both the numerator and denominator are functions of x. But if one part is a constant, there’s usually a quicker route.
| Situation | Better method |
|---|
| Both top & bottom are functions of x | quotient rule |
| Constant numerator, e.g. 2(3x−7)2 | rewrite as 2(3x−7)−2 → chain rule |
| Constant denominator, e.g. (3x−7)22 | treat 12 as a factor → chain rule |
The quotient rule
y = uv → dydx = vu′ − uv′v2
✓ given in the formula booklet
🧠 “Low d-high minus high d-low, over low squared”
“Low” = denominator v, “high” = numerator u. So: (low × derivative of high) − (high × derivative of low), all over low². The order — and the minus — are what keep you right.
Using the quotient rule
🧭 Recipe — quotient rule
- Identify u and v: u = numerator, v = denominator.
- Differentiate each to get u′ and v′ (chain rule may be needed).
- Apply the formula vu′ − uv′v2 — mind the order and the minus.
- Simplify if straightforward or if the question requires a particular form.
The “square” tip: lay u, v, u′, v′ out in a 2×2 grid. The pairs sit on opposite diagonals (v with u′, u with v′) — but unlike the product rule, the order is fixed: vu′ must come before uv′.
🤔 Quotient as a disguised product
Any fraction uv can be rewritten as u(v)−1 — a product — and differentiated with the product rule plus chain rule instead. The quotient rule is just the tidied-up result of doing exactly that, so the two methods always agree. Use whichever you find cleaner; the quotient rule usually saves a step.
Worked examples
WE 1Differentiate f(x) = cos 2x3x + 2
Both top and bottom are functions of x. Chain rule needed on cos 2x.
u = cos 2x, v = 3x + 2
u′ = −2 sin 2x (chain rule), v′ = 3
f′(x) = (3x + 2)(−2 sin 2x) − (cos 2x)(3)(3x + 2)²
f′(x) = −2(3x + 2)sin 2x − 3 cos 2x(3x + 2)²
Nothing obvious to simplify, and no particular form was required.
WE 2Differentiate y = xx + 1
A simple rational function — both parts functions of x.
u = x, v = x + 1
u′ = 1, v′ = 1
y′ = (x + 1)(1) − (x)(1)(x + 1)² = x + 1 − x(x + 1)²
y′ = 1(x + 1)²
WE 3Differentiate y = exx
Exponential over x.
u = eˣ, v = x
u′ = eˣ, v′ = 1
y′ = x·eˣ − eˣ·1x²
y′ = eˣ(x − 1)x²
WE 4Differentiate y = 2xsin x
Both functions of x; mind the order in the numerator.
u = 2x, v = sin x
u′ = 2, v′ = cos x
y′ = sin x·2 − 2x·cos xsin²x
y′ = 2 sin x − 2x cos xsin²x
WE 5Differentiate y = 2(3x − 7)2 — the smarter way
Constant numerator: rewrite with a negative power and use the chain rule instead of the quotient rule.
y = 2(3x − 7)⁻²
y′ = 2 · (−2)(3x − 7)⁻³ · 3 (chain rule)
y′ = −12(3x − 7)³
The quotient rule would work too, but this is quicker.
💡 Top tips
- Order is everything — vu′ comes first, then subtract uv′.
- Use the square layout and label u, v, u′, v′ clearly.
- Constant on top or bottom? Rewrite with powers and use the chain rule — it’s faster.
- Chain rule inside — composite numerators/denominators need it (e.g. cos 2x → −2 sin 2x).
- Don’t over-simplify — if nothing cancels neatly and no form is asked for, leave it.
- Match the notation the question uses.
⚠ Common mistakes
- Getting the order wrong — swapping to uv′ − vu′ flips every sign.
- Forgetting to square the denominator — it’s v2, not v.
- Forgetting the chain rule inside — differentiate composite u or v fully.
- Using it when you don’t need to — a constant top or bottom is quicker with powers + chain rule.
- Sign slips from the minus — the subtraction in the numerator trips people up; write it out carefully.
Next up — Related Rates of Change. You now have the full toolkit — chain, product and quotient rules — for differentiating almost anything. The next topic puts the chain rule to work in a new context: linking how two quantities change over time (like a balloon’s radius and its volume), connecting their rates through a shared variable, usually t.
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