Product and Quotient Rules NCEA Practice Questions | Calc.nz

Skill overview

What these questions practise

The matched methods cover both product-rule terms and the ordered quotient-rule numerator. The resulting derivatives are used for tangents, normals, rates, stationary points, and proofs.

Relevant standards: NCEA Level 3 Differentiation — AS91578

This collection contains 30 matching walkthroughs across 10 examination years.

The grouping is based on the method recorded for each walkthrough. Use the linked NZQA standard and paper material for authoritative wording, diagrams, and assessment information.

Before you start

Common mistakes in these methods

Keep both product-rule terms: differentiate one factor at a time while leaving the other factor unchanged.
Keep the quotient-rule numerator in the correct order and square the original denominator before simplifying.
A factor can still be composite, so apply its inner derivative inside the product or quotient rule.

Related practice

Product and Quotient Rules questions from different years

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NCEA Level 3 · AS91578 2025 Differentiation 3 questions

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Question 1(c) — Product Rule Product rule and the gradient of a normal. Question 2(b) — Product Rule Product rule and deciding whether a model is increasing or decreasing. Question 2(c) — Quotient Rule Quotient rule and tangents with a given gradient.

NCEA Level 3 · AS91578 2024 Differentiation 3 questions

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Question 1(b) — Product Rule Product rule differentiation and evaluating the gradient at x = 0. Question 3(a) — Chain Rule Product rule with square root of x and a trig chain rule. Question 3(c) — Quotient Rule Quotient rule differentiation and finding stationary points.

NCEA Level 3 · AS91578 2023 Differentiation 3 questions

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Question 1(b) — Product Rule Product rule differentiation and evaluating a rate of change. Question 2(a) — Product Rule Rewriting x squared divided by cos x and using the product rule. Question 2(c) — Quotient Rule Quotient rule differentiation and horizontal tangents.

NCEA Level 3 · AS91578 2022 Differentiation 2 questions

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Question 2(a) — Chain Rule Product rule with a trig chain rule. Question 2(c) — Quotient Rule Quotient rule and finding when motion is stationary.

NCEA Level 3 · AS91578 2021 Differentiation 7 questions

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Question 1(a) — Chain Rule Product rule with exponential and trigonometric chain rules. Question 1(c) — Chain Rule Product and chain rules for stationary points. Question 2(b) — Quotient Rule Quotient rule and stationary points. Question 2(c) — Product Rule Product rule and finding the equation of a normal. Question 3(a) — Quotient Rule Quotient rule with cot x and csc squared x. Question 3(d) — Quotient Rule Quotient rule with a parameter and a given tangent gradient. Question 3(e) — Chain Rule Lamp and table optimisation using product and chain rules.

NCEA Level 3 · AS91578 2020 Differentiation 4 questions

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Question 1(c) — Quotient Rule Quotient rule and a stationary point on a logarithmic function. Question 1(d) — Product Rule Product rule and proving a tangent equation. Question 2(a) — Quotient Rule Quotient rule differentiation of a trigonometric fraction. Question 2(c) — Chain Rule Product and chain rules for stationary points.

NCEA Level 3 · AS91578 2019 Differentiation 2 questions

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Question 1(c) — Quotient Rule Quotient rule differentiation and tangent gradient. Question 1(d) — Product Rule Product rule and finding where a function is decreasing.

NCEA Level 3 · AS91578 2018 Differentiation 3 questions

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Question 2(d) — Product Rule Product rule and stationary points of an exponential function. Question 3(a) — Quotient Rule Quotient rule with an exponential numerator. Question 3(d) — Product Rule Product rule and the equation of a tangent.

NCEA Level 3 · AS91578 2017 Differentiation 1 question

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Question 3(a) — Chain Rule Product and chain rules for a logarithmic function.

NCEA Level 3 · AS91578 2016 Differentiation 2 questions

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Question 2(a) — Chain Rule Product and chain rules with a logarithm. Question 3(d) — Quotient Rule Quotient rule proof with exponential and trigonometric functions.

Keep practising

Chain rule Stationary points and optimisation

Page updated 9 August 2026.