Level 3 Differentiation Walkthrough

2021 NCEA Level 3 Differentiation Question 3(d)

2021 Paper

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Question

A curve has the equation

\[ y=\frac{4x+k}{4x-k}, \]

where \(k\) is a constant and \(x\ne\frac{k}{4}\).

The point \(P\) lies on the curve and has an \(x\)-coordinate of \(3\).

The gradient of the tangent to the curve at \(P\) is \(-\frac{8}{27}\).

Find the possible value(s) of \(k\).

First walkthrough idea

Focus to try first

Differentiate in terms of \(k\), then substitute \(x=3\) and the given gradient.

Step 1

Differentiate in terms of k

Treat \(k\) as a constant.

Show the first step’s working
\[ \frac{dy}{dx} = \frac{4(4x-k)-4(4x+k)}{(4x-k)^2} \] \[ \frac{dy}{dx} = \frac{-8k}{(4x-k)^2} \]

Walkthrough overview

What this question practises

This 2021 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Quotient rule with a parameter and a given tangent gradient.

This is Question 3(d) from the 2021 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 quotient rule with a parameter and a given tangent gradient. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Use brackets carefully and retain the squared denominator when applying the quotient rule.

Continue practising

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