AS 91578 · 2024 paper · derivatives & tangents

Question 1(a) Achieved

Practises The rules & standard derivatives

A chain-rule root, a product-rule tangent, a decreasing-interval question mixing a square with a log, stationary points of an exponential product, and a tangent at a point of inflection.

Differentiate f(x) = √(4 − 9x⁴). You do not need to simplify your answer.

The 2024 paper, page 2 ↗ Marking schedule, page 1 ↗

Work it out

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

1 Write the square root as a power.
f(x) = (4 − 9x⁴)^1/2
2 Chain rule: what are the outside and the inside, and what are their derivatives?
Outside: u^1/2, which differentiates to (1/2) u^−1/2.
Inside: u = 4 − 9x⁴, which differentiates to −36x³.
3 Multiply them.
f'(x) = (1/2)(4 − 9x⁴)^−1/2 × (−36x³) = −18x³(4 − 9x⁴)^−1/2

Answer f'(x) = −18x³(4 − 9x⁴)^−1/2, which is also (−18x³)/(√(4 − 9x⁴)).

What the marker wanted

Achieved The correct derivative, in any equivalent form.

In our words, from NZQA’s 2024 marking schedule.

How did it go?

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