Level 3 Differentiation Walkthrough

2018 NCEA Level 3 Differentiation Question 2(d)

2018 Paper

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Question

If

\[y=e^x(2x^2-x-1),\]

find the value(s) of \(x\) for which \(\frac{dy}{dx}=0\).

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 product rule, factor the resulting quadratic, and remember that \(e^x\) is never zero.

Step 1

Differentiate using the product rule

Differentiate each factor once.

Show the first step’s working
\[\frac{dy}{dx}=e^x(2x^2-x-1)+e^x(4x-1)\] \[\frac{dy}{dx}=e^x(2x^2+3x-2)\]

Walkthrough overview

What this question practises

This 2018 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Product rule and stationary points of an exponential function.

This is Question 2(d) from the 2018 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 product rule and stationary points of an exponential function. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Differentiate both factors in turn and keep both product-rule terms.

Continue practising

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