Product, quotient & chain rules
Three rules cover almost every derivative you will meet. The product rule is (uv)′ = u′v + uv′; the quotient rule is (u/v)′ = (u′v − uv′)/v², where the order in the numerator matters; and the chain rule is (f(g(x)))′ = f′(g(x))·g′(x). Recognising which rule applies — and in what order for nested expressions — is most of the skill.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Identify the outermost structureIs the whole expression a product, a quotient, or a composite? That decides the first rule to apply.
- Product rule — differentiate each in turnu′v + uv′. Symmetric, so order does not matter.
- Quotient rule — keep the order(u′v − uv′)/v². Reversing the numerator flips the sign of the whole answer.
- Chain rule — work outside inDifferentiate the outer function at the inner one, then multiply by the inner derivative. Repeat for each layer.
Worked example
Differentiate y = x² sin(3x).
- This is a product with u = x² and v = sin(3x).
- u′ = 2x.
- v′ needs the chain rule: the derivative of sin(3x) is cos(3x)·3 = 3cos(3x).
- Apply the product rule: y′ = 2x·sin(3x) + x²·3cos(3x).
Answer. y′ = 2x sin(3x) + 3x² cos(3x).
Where marks get dropped
These are the specific errors that cost credit on product, quotient & chain rules questions — QED's rubric penalises each of them separately.
- Forgetting the inner derivative in the chain rule. The derivative of sin(3x) is 3cos(3x), not cos(3x) — the missing factor of 3 is the single most common error.
- Reversing the quotient rule numerator, which negates the answer.
- Using the product rule where a simple power rule suffices. Expand x²·x³ to x⁵ first when you can.
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Product, quotient & chain rules — frequently asked questions
How do I handle a triple product?
Apply the product rule twice, or use (uvw)′ = u′vw + uv′w + uvw′ — differentiate each factor in turn and add.
Can I avoid the quotient rule?
Yes, by writing u/v as u·v⁻¹ and using the product and chain rules. Many people find this less error-prone.
How deep can the chain rule nest?
Arbitrarily. For sin(cos(x²)) you multiply three derivatives: cos(cos(x²)) × (−sin(x²)) × 2x, working strictly from outside in.
The rest of Calculus
Limits, derivatives, integrals and their applications. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Limits & continuity
- 2Product, quotient & chain rules
- 3Tangent lines & linear approximation
- 4Extrema & optimisation
- 5Indefinite & definite integrals
- 6Substitution & integration by parts
- 7Series & convergence basics
- 8Implicit differentiation & related rates
- 9Mean value & intermediate value theorems
- 10Improper integrals & convergence
- 11Taylor & Maclaurin polynomials
- 12L’Hôpital’s rule & indeterminate forms
- 13Curve sketching: asymptotes, concavity & inflection
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