Level 3 Differentiation Walkthrough

2017 NCEA Level 3 Differentiation Question 1(b)

2017 Paper

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Question

Find the gradient of the tangent to the curve

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

at the point where \(x=0\).

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

First walkthrough idea

Focus to try first

Use the quotient rule, taking care to differentiate the exponential numerator with the chain rule before evaluating the gradient.

Step 1

Set up the quotient rule

Name the numerator and denominator so that each derivative is clear.

Show the first step’s working

For \(y=u/v\),

\[\frac{dy}{dx}=\frac{u'v-uv'}{v^2}.\]

Here

\[u=e^{2x},\qquad v=x+2.\]

Walkthrough overview

What this question practises

This 2017 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Quotient and chain rules, then evaluating a gradient.

This is Question 1(b) from the 2017 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 and chain rules, then evaluating a gradient. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Do not stop after differentiating the outside function; include the derivative of the inside function as a factor.

Continue practising

← Question 1(a) Question 1(c) →