2026 Prediction Examination
Level 3 Differentiation
91578 Apply differentiation methods in solving problems
Credits: Six
| Achievement | Achievement with Merit | Achievement with Excellence |
|---|---|---|
| Apply differentiation methods in solving problems. | Apply differentiation methods, using relational thinking, in solving problems. | Apply differentiation methods, using extended abstract thinking, in solving problems. |
Instructions
- Suggested time: 60 minutes.
- You should attempt ALL the questions in this paper.
- Show ALL relevant working.
- A calculator is permitted; a graphics calculator is best.
- The formula sheet opens from the margin beside any question.
- Use the margin beside each question for its timer and worked answer.
- Start the timer below when you are ready to begin.
Main Exam Timer
Our 2026 prediction paper, written from the 2022 to 2025 papers and this year's assessment specification. Every question has a worked answer.
A Maths Study Hub practice paper. Not an official NZQA examination.
QUESTION ONE
(a)
Differentiate $f(x)=\mathrm{e}^{\cos 2x}$.
You do not need to simplify your answer.
(b)
Find the gradient of the tangent to the curve $y=\dfrac{x}{\sqrt{x+3}}$ at the point $(6,2)$.
You must use calculus and show any derivatives that you need to find when solving this problem.
(c)
Find the coordinates of any stationary points on the graph of
$$f(x)=\sin^2x+\cos x$$for $0\lt x\lt 2\pi$, and determine their nature.
You must use calculus and show any derivatives that you need to find when solving this problem.
(d)
The tangent to the curve $y=\ln(2x)$ at the point P passes through the origin, as shown below.
Find the equation of this tangent.
You must use calculus and show any derivatives that you need to find when solving this problem.
(e)
A curve is defined by the parametric equations
$$x=4\cos^3\theta\quad\text{and}\quad y=4\sin^3\theta,$$where $0\lt \theta\lt \frac{\pi}{2}$.
The tangent to the curve at the point P meets the $x$-axis at A and the $y$-axis at B.
Prove that the length AB is the same wherever P is on the curve.
You must use calculus and show any derivatives that you need to find when solving this problem.
QUESTION TWO
(a)
Differentiate $y=\dfrac{x^2+1}{\ln(3x)}$.
You do not need to simplify your answer.
(b)
After a bungy jump, the jumper bounces up and down. The jumper’s height above the river, $h$ metres, is modelled by
$$h=22+18\mathrm{e}^{-0.25t}\cos(1.8t)$$where $t$ is the time in seconds, $t\ge0$.
Find the velocity of the jumper when $t=2$, and state whether the jumper is moving up or down at that moment.
You must use calculus and show any derivatives that you need to find when solving this problem.
(c)
A water trough is 200 cm long. Its ends are isosceles triangles, 60 cm across the top and 40 cm deep, with the vertex at the bottom. The diagrams show the trough and a cross-section of it.
Water flows into the trough at a constant rate of 12 000 cm3 per minute.
At what rate is the depth of the water increasing when the water is 25 cm deep?
You must use calculus and show any derivatives that you need to find when solving this problem.
(d)
A curve is defined by the parametric equations
$$x=t^3-3t\quad\text{and}\quad y=t^2.$$Find the values of $t$ for which the curve is concave up.
You must use calculus and show any derivatives that you need to find when solving this problem.
(e)
After predators are removed from an island, a population of a rare bird is released there. The number of birds, $N$, $t$ years after the release, is modelled by
$$N=\frac{400}{1+19\mathrm{e}^{-0.5t}}$$Find when the population is increasing at its greatest rate, and how fast it is increasing then.
Show that, at that time, the population is half its long-term size.
You can assume that the rate you find is a maximum.
You must use calculus and show any derivatives that you need to find when solving this problem.
QUESTION THREE
(a)
Differentiate $y=x^3\cot(2x)$.
You do not need to simplify your answer.
(b)
The graph below shows the function $y=f(x)$.
For the function above:
(i) Find the value(s) of $x$ where $f(x)$ is continuous but not differentiable.
(ii) Find the value(s) of $x$ where $f'(x)=0$ and $f''(x)\gt 0$.
(iii) What is the value of $\displaystyle\lim_{x\to-1}f(x)$? State clearly if the value does not exist.
(c)
A curve is defined by the parametric equations
$$x=\mathrm{e}^{t}+\mathrm{e}^{-t}\quad\text{and}\quad y=\mathrm{e}^{t}-3\mathrm{e}^{-t}.$$Find the equation of the normal to the curve at the point $\left(\frac52,\frac12\right)$.
You must use calculus and show any derivatives that you need to find when solving this problem.
(d)
Show that $y=\sec(2x)$ satisfies the equation
$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=8y^3-4y.$$You must use calculus and show any derivatives that you need to find when solving this problem.
(e)
Isla is training for a kayak race on a river that flows at a constant speed of $c$ km h−1. She paddles 5 km upstream, from S to T, then turns and paddles 5 km back downstream to S.
She paddles at the same speed for the whole trip: $v$ km h−1 relative to the water, where $v\gt c$. The energy she uses each hour is $0.2v^3$ units.
Find the value of $v$, in terms of $c$, that makes the total energy for her trip as small as possible, and show that this least energy is $8c^2$ units.
You can assume that the value of $v$ you find gives the least energy.
You must use calculus and show any derivatives that you need to find when solving this problem.
How we chose these questions
A prediction paper is our best judgement of what the 2026 examination is likely to ask. It isn’t a copy of any past paper, and nobody can know the real questions in advance. We built this one from NZQA’s 2022 to 2025 Differentiation examinations and marking schedules, the examiners’ reports on those years, and the 2026 assessment specification, and we checked every paper back to 2013 so that no question repeats one.
Every question is here for at least one of three reasons
- It’s in the 2026 specificationThe specification says problems will test understanding of the concepts of differentiation, every derivative must be shown, candidates may have to form their own equations for Merit and Excellence, and equations of tangents may be included.
- It comes up every yearEvery paper since 2013 has had related rates of change, a curve defined by parametric equations, and an optimisation problem, and every one except 2020 has had a graph to read. Tangents and normals are in every paper. All of them are here.
- It’s where marks are lostThe examiners’ reports name the mistakes candidates make most often: losing a factor in the chain rule, forgetting to multiply by dt/dx in a parametric second derivative, using a point’s coordinates as the value of t, and struggling when a question joins calculus to coordinate geometry. We built these into the questions.
Question by question
| Question | What it tests | Why it’s here |
|---|---|---|
| 1(a) A | Differentiating e to the power of a trigonometric function | The 2021 report said the chain rule caused “numerous careless errors”, such as losing the 2 from sin 2x. This needs the chain rule twice. |
| 1(b) A | The gradient of a tangent, for a quotient with a square root in it | A gradient at a point is in nearly every paper. The 2024 and 2025 reports credit candidates who write surds as powers before differentiating. |
| 1(c) M | Stationary points of a trigonometric function, and their nature | Stationary points and their nature were Merit parts in 2023 and 2024. The 2025 report says candidates must set the whole derivative equal to zero; here, cancelling a factor loses a stationary point. |
| 1(d) M | The tangent to a logarithmic curve that passes through the origin | The 2021 report said “few candidates knew how to solve a problem involving a tangent through a point not on the given curve” and that it “needs to be included in all teaching programmes”. |
| 1(e) E | Tangents to a parametric curve that always cut off the same length | The reports from 2022 to 2025 say candidates struggle when calculus meets coordinate geometry, and the 2025 report singled out derivatives that mix powers and trigonometry. This combines both, in a proof. |
| 2(a) A | The quotient rule, with a logarithm in the denominator | Most questions in NZQA’s papers open with a derivative to find. The reports warn candidates not to simplify when they are told they needn’t. |
| 2(b) A | The velocity of a bungy jumper | A rate of change in context is in many papers, including two in 2025. The 2024 report noted calculator errors with trigonometric and exponential functions; this one needs radians. |
| 2(c) M | Related rates: water flowing into a trough | Related rates is in every paper since 2013. Like the 2024 cone of flour, the volume has to be written in one variable first. |
| 2(d) M | Where a parametric curve is concave up | The 2025 report said candidates found parametric second derivatives by “omitting to multiply by dt/dx”. Here that mistake gives the wrong answer. |
| 2(e) E | When a bird population grows fastest | Inflection points were the Excellence theme in 2022, 2024 and 2025. Here the inflection has a meaning: the moment the population grows fastest, which turns out to be at half its long-term size. |
| 3(a) A | The product rule with a reciprocal trigonometric function | Reciprocal trigonometric functions are in the standard and the formulae booklet, and in recent papers (cot in 2021 and 2023, sec in 2024 and 2025). |
| 3(b) A | Reading a graph: continuity, differentiability, gradients and a limit | A graph like this is in every paper except 2020, and every report from 2021 to 2025 lists misreading it as a reason for Not Achieved. |
| 3(c) M | The normal to a parametric curve at a given point | The 2021 report described candidates substituting a coordinate as the value of t. Normals are in most papers. |
| 3(d) M | Showing that a function satisfies an equation with its second derivative | An equation like this is in 8 of the 13 papers since 2013, most recently 2024, whose report said the product rule inside a second derivative caused errors. |
| 3(e) E | The best speed for a kayak trip up a river and back | An optimisation is in every paper, and the reports from 2022 to 2025 say forming the model is the step candidates find hardest. This model has to be built from the speeds. |
None of the Excellence questions, 1(e), 2(e) and 3(e), repeats an NZQA question from 2013–2025.
NZQA sources: 2026 assessment specification, 2025 assessment report, 2025 examination, 2025 assessment schedule. The 2022 to 2024 papers, schedules and reports are on the same NZQA pages.
Maths Study Hub is independent and is not affiliated with or endorsed by NZQA. Written October 2026.