Level 3 Differentiation Walkthrough

2017 NCEA Level 3 Differentiation Question 3(a)

2017 Paper

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Question

\[\text{Differentiate }y=x\ln(3x-1).\]

You do not need to simplify your answer.

First walkthrough idea

Focus to try first

Treat \(x\) and \(\ln(3x-1)\) as two factors. The logarithm also needs the chain rule.

Step 1

Identify the two rules

The expression is a product, and the logarithm contains the inner function \(3x-1\).

Show the first step’s working

Let \(u=x\) and \(v=\ln(3x-1)\). The product rule is

\[\frac{d}{dx}(uv)=u'v+uv'.\]

For real values, the original function is defined when \(3x-1>0\), so \(x>\tfrac13\).

Walkthrough overview

What this question practises

This 2017 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Product and chain rules for a logarithmic function.

This is Question 3(a) from the 2017 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 product and chain rules for a logarithmic function. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Do not stop after differentiating the outside function; include the derivative of the inside function as a factor.

Continue practising

← Question 2(e) Question 3(b) →