Level 3 Differentiation Walkthrough

2023 NCEA Level 3 Differentiation Question 2(c)

2023 Paper

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Question

\[ f(x)=\frac{e^x}{x^2+2x} \]

Find the \(x\)-value(s) of any point(s) on the curve where the tangent to the curve is parallel to the \(x\)-axis.

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

First walkthrough idea

Hint to try first

A tangent parallel to the \(x\)-axis means \(f'(x)=0\).

Step 1

Simplify the derivative

After using the quotient rule, the numerator simplifies to \(e^x(x^2-2)\).

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

After using the quotient rule, the numerator simplifies to \(e^x(x^2-2)\).

Key result

\[ f'(x)=\frac{e^x(x^2-2)}{(x^2+2x)^2} \]

Walkthrough overview

What this question practises

This 2023 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Quotient rule differentiation and horizontal tangents.

This is Question 2(c) from the 2023 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 horizontal tangents. 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(b) Question 2(d) →