← Differentiation methods: past papers

AS 91578 · 2025 paper · the graph, related rates, rectangle optimisation

Question 3(b) Achieved

Practises Tangents, normals and curve features

Reading differentiability, stationary points and a limit straight off a drawn graph, then a log quotient, a related-rates sphere, a parametric gradient, and a rectangle inscribed in a curve.

A curve has equation y = (ln x)/2x. Find the x-coordinate(s) of any stationary point(s) on the curve.

The 2025 paper, page 10 ↗ Marking schedule, page 5 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

  1. 1 Stationary points are where dy/dx = 0. Which rule differentiates (ln x)/2x?

    The quotient rule, with u = ln x, u' = 1/x, v = 2x, v' = 2.

  2. 2 Differentiate and simplify.

    dy/dx = (1/x · 2x − 2 ln x)/(2x)2 = (2 − 2 ln x)/4x2

  3. 3 Set it to zero. When is a fraction equal to zero?

    When its numerator is: 2 − 2 ln x = 0, so ln x = 1 and x = e ≈ 2.718.
    The denominator 4x2 cannot be zero, because ln x needs x > 0.
Answer x = e ≈ 2.718.

What the marker wanted

  • Achieved x = e, from a correct derivative.

In our words, from NZQA’s 2025 marking schedule.

✕

Common mistake

Crossing out the denominator before writing that dy/dx = 0. NZQA’s 2025 report names this part: write the whole derivative equal to zero first, so the marker can see every step.

How did it go?

Got it moves it further out. Review brings it back tomorrow.