← Differentiation methods: past papers

AS 91578 · 2025 paper · logs, exponentials, tangents, radius of curvature

Question 2(a) Achieved

Practises The rules & standard derivatives

A reciprocal minus a log, a product that has to be differentiated to decide increasing or decreasing, two tangents of the same gradient to a rational curve, a parametric tangent in sec and tan, and a given curvature formula to apply.

Differentiate f(x) = 7/x − ln(3x3 − 2x2 + 5). You do not need to simplify your answer.

The 2025 paper, page 5 ↗ Marking schedule, page 3 ↗

Work it out

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

  1. 1 Take the two terms one at a time. How do you write 7/x so the power rule applies?

    7/x = 7x−1, which differentiates to −7x−2 = −7/x2.

  2. 2 What is the derivative of ln of a function?

    The derivative of the inside over the inside:
    (d/dx) ln(3x3 − 2x2 + 5) = (9x2 − 4x)/(3x3 − 2x2 + 5)

  3. 3 Put the two together. Keep the minus sign between them.

    f'(x) = −7/x2 − (9x2 − 4x)/(3x3 − 2x2 + 5)
Answer f'(x) = −7/x2 − (9x2 − 4x)/(3x3 − 2x2 + 5), with no need to simplify.

What the marker wanted

  • Achieved A correct derivative, in any form.

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

How did it go?

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