NCEA 91578: differentiation (Level 3 Calculus)

What NCEA 91578 assesses

NZQA says you are assessed on applying differentiation methods in solving problems, and that the problems are set in real life or mathematical contexts. The standard (version 2) lists the methods:

• derivatives of power, exponential and logarithmic (base e only) functions
• derivatives of trigonometric functions, including the reciprocal functions
• the chain, product and quotient rules
• derivatives of parametric functions
• equations of normals, and maxima, minima and points of inflection
• optimisation, and related rates of change
• properties of graphs: limits, differentiability, continuity and concavity

The three grades read the same way in the standard, with a different kind of thinking. Achieved applies differentiation methods in solving problems. Merit uses relational thinking, which means selecting and carrying out a logical sequence of steps, connecting concepts or representations, and relating findings to a context. Excellence uses extended abstract thinking, which means devising a strategy, developing a chain of logical reasoning or proof, or forming a generalisation.

The NCEA 91578 exam in 2026 and 2027

NCEA 91578 is one of three Level 3 Calculus externals, with 91577 (complex numbers) and 91579 (integration). The 2026 Level 3 Calculus exam is on Monday 16 November, in the morning session. NZQA's 2027 timetable is not out yet: it says exams run from 9 November to 3 December 2027 and the subject timetable is due in December 2026. NZQA's assessment specifications for both years say the exam is on printed paper, 91578 is still an external standard worth 6 credits, and a Formulae and Tables Booklet is provided.

• Calculator: bring an approved calculator, preferably a graphing calculator. NZQA says candidates without one will be disadvantaged.
• Show the derivative: you must show any derivatives you need. A correct answer from a graphing calculator gets no credit unless you also give the derived function.
• Optimisation: at Achieved and Merit you are not asked to prove that a solution is a maximum or a minimum. The question tells you to assume it, so a proof can be asked for at Excellence.

How NCEA 91578 is marked: grade score marking

Each question gets a score from N0 to E8, and the three scores are added to give a total out of 24. Evidence from your working is counted in units: Achieved evidence (u), Merit evidence (r) and Excellence evidence (t). In NZQA's schedules, N1 is one partial solution, N2, A3 and A4 come from one, two and three units of u, M5 and M6 from one and two units of r, and E7 and E8 from one and two units of t. A total is then turned into a grade using the cut scores for that year.

NCEA 91578 cut scores

These cut scores are NZQA's grade boundaries for the exam. They change a little each year, because the papers differ. Over these five years Achieved needed 7 or 8, Merit 13 or 14, and Excellence 19 or 20. The 2026 cut scores will appear in NZQA's 2026 assessment schedule once the exam has been marked.

Cut scores 2025

From NZQA's 2025 assessment schedule, out of 24: Not Achieved 0 to 7, Achieved 8 to 13, Merit 14 to 19, Excellence 20 to 24.

Cut scores 2024

From NZQA's 2024 assessment schedule, out of 24: Not Achieved 0 to 7, Achieved 8 to 12, Merit 13 to 18, Excellence 19 to 24.

Cut scores 2023

From NZQA's 2023 assessment schedule, out of 24: Not Achieved 0 to 7, Achieved 8 to 12, Merit 13 to 18, Excellence 19 to 24.

Cut scores 2022

From NZQA's 2022 assessment schedule, out of 24: Not Achieved 0 to 7, Achieved 8 to 13, Merit 14 to 19, Excellence 20 to 24.

Cut scores 2021

From NZQA's 2021 assessment schedule, out of 24: Not Achieved 0 to 6, Achieved 7 to 12, Merit 13 to 18, Excellence 19 to 24.

Worked examples at each grade

Achieved: product rule with a logarithm

Differentiate f(x) = x² ln(3x) and find f′(2). The product rule gives:

So f′(2) = 4 ln 6 + 2 = 9.17 to 3 significant figures. Check the result in the Derivative Calculator.

Merit: related rates

An oil slick spreads as a circle, and its area grows at 12 m² per minute. How fast is the radius increasing when the radius is 20 m? The area and the radius are linked, so differentiate with respect to time:

Merit asks you to link the steps and say what the answer means: at that moment the radius is growing at about 0.0955 metres per minute. See 2πr at r = 20 in the Derivative Calculator.

Excellence: optimisation with a proof

An open-topped box has a square base of side x cm and a volume of 500 cm³. Find the side that minimises the surface area, and prove it is a minimum. The height is 500 ÷ x², so the surface area is:

Excellence rewards forming the model yourself and justifying the result with the second derivative. The Derivative Calculator shows the gradient is 0 at x = 10.

Now classify some stationary points in our Calculus course, then press Mark to check your answer in the course:

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. We do not publish official or past paper answers. Every question on the daily paper is our own.

Use the daily paper as a practice exam

Uni-Sine publishes a free daily paper on the NCEA Level 3 Calculus track. It is a short timed paper of 8 questions worth 25 to 35 marks, not a full NCEA exam, and it is not written by NZQA. It covers the chain, product and quotient rules, tangents and normals, stationary points, rates of change and parametric differentiation, which is most of 91578. It also includes differentiation from first principles, which is not on the standard's list, so treat that as extension. For the other two externals and the credits and endorsements, see the NCEA Level 3 Calculus guide.

Is NCEA 91578 changing?

NCEA still applies in 2026 and 2027, and NZQA's 2027 specification for Level 3 Calculus keeps 91578 as a 6-credit external. The Ministry of Education says a new New Zealand Advanced Certificate of Education (NZACE) for Year 13 begins in 2030. Check NZQA before planning beyond 2027.

Common mistakes in NCEA 91578

• Relying on the calculator. A correct answer with no derived function earns no credit.
• Calculator errors with radians. NZQA's 2025 assessment report says many candidates made them when substituting radian values into trigonometric functions.
• Not answering the question. The 2025 report gives the example of solving for k in terms of x when the question asked for x in terms of k.

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