Level 3 Differentiation Walkthrough

2025 NCEA Level 3 Differentiation Question 3(d)

2025 Paper

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Question

The equation of a curve is given by the pair of parametric equations \[ x=\frac{5}{e^{2t}} \quad \text{and} \quad y=5e^{2t}. \]

Find the coordinates of the point(s) on the curve where the gradient is \(-2\).

You must use calculus and show any derivatives that you need to find when solving this problem.

First walkthrough idea

Hint to try first

Rewrite \(x=\frac{5}{e^{2t}}\) as \(x=5e^{-2t}\) before differentiating.

Step 1

Find \(\frac{dy}{dx}\)

The gradient simplifies to \(-e^{4t}\).

Show the first step’s working

The gradient simplifies to \(-e^{4t}\).

Key result

\[ -e^{\left(4 t\right)} \]

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Solving a parametric gradient condition.

This is Question 3(d) from the 2025 NCEA Level 3 Differentiation paper for AS91578 — Apply differentiation methods in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91578.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise solving a parametric gradient condition. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

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