Level 3 Differentiation Walkthrough

2022 NCEA Level 3 Differentiation Question 2(b)

2022 Paper - Tangent gradient using the chain rule

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Question

Find the gradient of the tangent to the curve \(y=(3x^2-2)^3\) when \(x=2\).

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

First walkthrough idea

Tip to try first

Treat the expression as a power on the outside and a quadratic on the inside.

Step 1

Identify the differentiation rule

Reveal the explanation and working for this step.

Show the first step’s working

Use the chain rule because the expression is a function raised to a power.

Key result

Chain rule

This is a composition of functions, so the chain rule is the right starting point.

Walkthrough overview

What this question practises

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

Method: Chain rule and finding a tangent gradient.

This is Question 2(b) 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 chain rule and finding a tangent gradient. 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(a) Question 2(c) →