← Differentiation methods: past papers

AS 91578 · 2023 paper · trig, quotients, the helicopter

Question 2(b) Achieved

Practises The rules & standard derivatives

A quotient with cos, a cot gradient, a stationary point on an exponential quotient, inflection on x2 ln x, and a related-rates chase where the rate given is an angle.

Find the gradient of the tangent to the curve y = cot(2x) at the point where x = π/12.

The 2023 paper, page 6 ↗ Marking schedule, page 4 ↗

Work it out

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

  1. 1 What is the derivative of cot? Then what does the 2x add?

    (d/du) cot u = −cosec2 u, and the chain rule brings out a 2:
    dy/dx = −2cosec2(2x) = (−2)/(sin2(2x))

  2. 2 Put x = π/12 in.

    2x = π/6 and sinπ/6 = 1/2, so dy/dx = (−2)/(1/4) = −8.
Answer The gradient is −8.

What the marker wanted

  • Achieved The correct derivative, and a gradient of −8.

In our words, from NZQA’s 2023 marking schedule.

How did it go?

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