Level 3 Differentiation Walkthrough

2024 NCEA Level 3 Differentiation Question 3(c)

2024 Paper

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Question

Find the \(x\)-value(s) of any stationary point(s) on the graph of the function \[ f(x)=\frac{x^2-5x+4}{x^2+5x+4}. \]

You do not need to determine the nature of any stationary point(s) found.

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

First walkthrough idea

Hint to try first

Use the quotient rule here.

Step 1

Differentiate the quotient

After the quotient rule, the numerator simplifies a lot more nicely than it first looks.

Show the first step’s working
\[ f'(x)=\frac{(x^2+5x+4)(2x-5)-(x^2-5x+4)(2x+5)}{(x^2+5x+4)^2} \]

After the quotient rule, the numerator simplifies a lot more nicely than it first looks.

Key result

\[ \frac{\left(10 x^{2} - 40\right)}{\left(x^{2} + 5 x + 4\right)^{2}} \]

Walkthrough overview

What this question practises

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

Method: Quotient rule differentiation and finding stationary points.

This is Question 3(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 quotient rule differentiation and finding stationary points. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Use brackets carefully and retain the squared denominator when applying the quotient rule.

Continue practising

← Question 3(b) Question 3(d) →