Level 3 Differentiation Walkthrough

2025 NCEA Level 3 Differentiation Question 2(d)

2025 Paper

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Question

Find the equation of the tangent to the graph defined by the pair of equations \[ x=2\sec t \quad \text{and} \quad y=5\tan t, \] at the point on the graph where \(t=\frac{\pi}{6}\).

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

First walkthrough idea

Hint to try first

For a parametric curve, \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\).

Step 1

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

Dividing \(\frac{dy}{dt}\) by \(\frac{dx}{dt}\) simplifies to \(\frac{5}{2\sin t}\).

Show the first step’s working

Dividing \(\frac{dy}{dt}\) by \(\frac{dx}{dt}\) simplifies to \(\frac{5}{2\sin t}\).

Key result

\[ \frac{5}{\left(2 \sin\left(t\right)\right)} \]

Walkthrough overview

What this question practises

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

Method: Parametric gradients and tangent equations.

This is Question 2(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 parametric gradients and tangent equations. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Use the derivative for the gradient and the original curve for the point before forming the tangent equation.

Continue practising

← Question 2(c) Question 2(e) →