Level 3 Differentiation Walkthrough

2024 NCEA Level 3 Differentiation Question 2(c)

2024 Paper

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Question

Show that \[ y=\sin(x^2)-\cos x \] is a solution to the equation \[ \frac{d^2y}{dx^2}+4x^2y=2\cos(x^2)+(1-4x^2)\cos x. \]

First walkthrough idea

Focus to try first

finding \(y'\) and \(y''\), then substituting back into the left-hand side and simplifying until the matching terms appear.

Step 1

Differentiate once

The first term needs the chain rule, and the second term differentiates to \(+\sin x\).

Show the first step’s working

The first term needs the chain rule, and the second term differentiates to \(+\sin x\).

Key result

\[ 2 x \cdot \cos\left(x^{2}\right) + \sin\left(x\right) \]

Walkthrough overview

What this question practises

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

Method: First and second derivatives in a proof-style differential-equation question.

This is Question 2(c) 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 first and second derivatives in a proof-style differential-equation question. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

← Question 2(b) Question 2(d) →