Level 3 Differentiation Walkthrough
2023 NCEA Level 3 Differentiation Question 1(c)
2023 Paper
Question
The graph shows the curve \(y=\frac{2}{(x+1)^3}\), along with the tangent to the curve drawn at \(x=1\).
A second tangent to this curve is drawn which is parallel to the first tangent shown.
Find the \(x\)-coordinate of the point where this second tangent touches the curve.
You must use calculus and show any derivatives that you need to find when solving this problem.
First walkthrough idea
Hint to try first
Parallel tangents have the same gradient.
Step 1
Differentiate the curve
You can get that from the chain rule or by using the quotient rule and simplifying.
Show the first step’s working
You can get that from the chain rule or by using the quotient rule and simplifying.
Key result
\[ y'=-\frac{6}{(x+1)^4} \]Walkthrough overview
What this question practises
This 2023 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.
Method: Using derivatives to find a parallel tangent on a rational curve.
This is Question 1(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 using derivatives to find a parallel tangent on a rational curve. Use the hints to plan the method, then repeat the question without hints and check each step.
Common mistake to avoid
Use the derivative for the gradient and the original curve for the point before forming the tangent equation.