← Differentiation methods: past papers

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

Question 2(a) 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.

Differentiate f(x) = x2/(cos x). You do not need to simplify your answer.

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 A quotient. What are u and v, and their derivatives?

    u = x2, u' = 2x; v = cos x, v' = −sin x.

  2. 2 Apply the quotient rule.

    f'(x) = (2x cos x − x2(−sin x))/(cos2 x) = (2x cos x + x2 sin x)/(cos2 x)
    (Writing it as x2 sec x and using the product rule gives 2x sec x + x2 sec x tan x: the same thing.)
Answer f'(x) = (2x cos x + x2 sin x)/(cos2 x).

What the marker wanted

  • Achieved The correct derivative.

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

How did it go?

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