Logarithms Worked Solution – 2018 NCEA Level 3 Q2(b) (AS91578)

Question

A particle is travelling in a straight line. The distance, in metres, travelled by the particle may be modelled by

\[s(t)=\ln(3t^2+3t+1),\qquad t\ge0,\]

where \(t\) is time measured in seconds. Find the velocity of this particle after \(2\) seconds.

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

First walkthrough idea

Focus to try first

Velocity is \(ds/dt\). Differentiate the logarithm using the chain rule, then evaluate at \(t=2\).

Step 1

Differentiate the distance function

For \(\ln u\), the derivative is \(u'/u\).

Show the first step’s working

v(t)=\frac{ds}{dt}=\frac{6t+3}{3t^2+3t+1}

Walkthrough overview

What this question practises

This 2018 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Finding velocity from a logarithmic distance function.

This is Question 2(b) from the 2018 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.

Page updated 9 August 2026.

Learning summary

Review the method, not only the answer

This walkthrough helps you practise finding velocity from a logarithmic distance function. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Keep logarithm domain restrictions and any inner-function factor visible throughout the working.

Continue practising

← Question 2(a) Question 2(c) →