Level 3 Differentiation Walkthrough

2022 NCEA Level 3 Differentiation Question 3(d)

2022 Paper - Stationary points and their nature

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Question

\[ y=9x-2+\frac{3}{3x-1} \]

Find the \(x\)-value(s) of any stationary point(s) on the graph and determine their nature.

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

First walkthrough idea

Tip to try first

Differentiate first, then set \(y'=0\) to find stationary points.

Step 1

Differentiate the function

Reveal the explanation and working for this step.

Show the first step’s working

Differentiate each term, using the chain rule on the fraction term.

Worked result

\[ 9 - \frac{9}{\left(3 x - 1\right)^{2}} \]

The derivative of \(\frac{3}{3x-1}\) is \(-\frac{9}{(3x-1)^2}\).

Walkthrough overview

What this question practises

This 2022 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Finding stationary points and classifying them.

This is Question 3(d) from the 2022 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 finding stationary points and classifying them. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

After solving the derivative condition, check the point's nature and answer the conclusion the question actually asks for.

Continue practising

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