Level 3 Differentiation Walkthrough

2016 NCEA Level 3 Differentiation Question 1(e)

2016 Paper

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Question

A curve is defined by the function

\[f(x)=e^{-(x-k)^2}.\]

Find, in terms of \(k\), the \(x\)-coordinate(s) for which \(f''(x)=0\).

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

First walkthrough idea

Focus to try first

Differentiate twice with the chain and product rules, then use the fact that the exponential factor is never zero.

Step 1

Find the first derivative

Apply the chain rule to the exponential.

Show the first step’s working
\[ \frac{d}{dx}\bigl(-(x-k)^2\bigr)=-2(x-k) \] \[ f'(x)=-2(x-k)e^{-(x-k)^2}=(2k-2x)e^{-(x-k)^2}. \]

Walkthrough overview

What this question practises

This 2016 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: First and second chain-rule derivatives of an exponential.

This is Question 1(e) from the 2016 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 first and second chain-rule derivatives of an exponential. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

← Question 1(d) Question 2(a) →