Level 3 Differentiation Walkthrough

2025 NCEA Level 3 Differentiation Question 2(c)

2025 Paper

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Question

There are two tangents to the curve \[ y=f(x)=\frac{x^2}{x+4} \] that have the equation tangents of the form \(y=-3x+c\).

y = f(x) y = f(x) x = -4 x y

Find the coordinates of the two points of contact of these tangents with the curve \(y=f(x)\).

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

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First walkthrough idea

Hint to try first

Differentiate \(f(x)=\frac{x^2}{x+4}\) using the quotient rule.

Step 1

Differentiate the curve

The quotient rule simplifies to \(\frac{x(x+8)}{(x+4)^2}\).

Show the first step’s working

The quotient rule simplifies to \(\frac{x(x+8)}{(x+4)^2}\).

Key result

\[ \frac{x \left(x + 8\right)}{\left(x + 4\right)^{2}} \]

Walkthrough overview

What this question practises

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

Method: Quotient rule and tangents with a given gradient.

This is Question 2(c) 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 quotient rule and tangents with a given gradient. 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

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