Level 3 Differentiation Walkthrough

2024 NCEA Level 3 Differentiation Question 2(b)

2024 Paper

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Question

An object is travelling in a straight line. Its displacement, in metres, is given by the formula \[ s(t)=\ln(3t^2+5t+2), \] where \(t>0\) and \(t\) is time, in seconds.

Find the velocity of this object when \(t=1\) second.

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

First walkthrough idea

Hint to try first

We are given displacement, so differentiate to get velocity.

Step 1

Spot the method

The logarithm is the outside function, and \(3t^2+5t+2\) is the inside.

Show the first step’s working

The logarithm is the outside function, and \(3t^2+5t+2\) is the inside.

Key result

Because the logarithm has a quadratic inside it

Walkthrough overview

What this question practises

This 2024 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Chain rule differentiation of a logarithmic displacement function.

This is Question 2(b) from the 2024 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 chain rule differentiation of a logarithmic displacement 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

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