Level 3 Differentiation Walkthrough
2024 NCEA Level 3 Differentiation Question 1(e)
2024 Paper
Question
A curve is defined by the equation \[ y=\frac{2x^2-1-2x\ln x}{x},\qquad x>0. \]
The curve has a point of inflection at the point \(P\).
Find the equation of the tangent to the curve at the point \(P\).
You must use calculus and show any derivatives that you need to find when solving this problem.
First walkthrough idea
Hint to try first
Start by finding where the second derivative is zero, because that tells us where \(P\) is.
Step 1
Find the first derivative
That is the first derivative we need before going on to the second derivative.
Show the first step’s working
That is the first derivative we need before going on to the second derivative.
Key result
Walkthrough overview
What this question practises
This 2024 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.
Method: Locating a point of inflection and finding the tangent there.
This is Question 1(e) from the 2024 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 locating a point of inflection and finding the tangent there. Use the hints to plan the method, then repeat the question without hints and check each step.
Common mistake to avoid
A zero second derivative alone is not enough; check the required change in concavity or other supporting evidence.