Level 3 Differentiation Walkthrough

2021 NCEA Level 3 Differentiation Question 2(b)

2021 Paper

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Question

A curve has the equation

\[ y=\frac{x^2}{x+1}. \]

Find the \(x\)-coordinate(s) of any stationary point(s) on the curve.

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

First walkthrough idea

Focus to try first

Use the quotient rule carefully, then set the numerator of the derivative equal to zero.

Step 1

Apply the quotient rule

Let the numerator be \(x^2\) and the denominator be \(x+1\).

Show the first step’s working
\[ \frac{dy}{dx} = \frac{2x(x+1)-x^2(1)}{(x+1)^2} \] \[ \frac{dy}{dx} = \frac{x^2+2x}{(x+1)^2} \]

Walkthrough overview

What this question practises

This 2021 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Quotient rule and stationary points.

This is Question 2(b) from the 2021 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 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 2(a) Question 2(c) →