Level 3 Differentiation Walkthrough

2025 NCEA Level 3 Differentiation Question 1(a)

2025 Paper

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Question

\[ \text{Differentiate } f(x)=(5x^3-2x+1)^5. \]

You do not need to simplify your answer.

First walkthrough idea

Tip to try first

Treat the bracket \(5x^3-2x+1\) as one inside function before you start differentiating.

Step 1

Recognise a composite function

Treat the whole bracket as one inside function.

Show the first step’s working

The expression has the form \(f(x)=\bigl(u(x)\bigr)^5\), where

\[ u(x)=5x^3-2x+1. \]

That means this is a chain rule question: differentiate the outside power, then multiply by the derivative of the inside.

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Chain rule differentiation of a composite polynomial.

This is Question 1(a) from the 2025 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 composite polynomial. 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 1(b) →