← Differentiation methods: past papers

AS 91578 · 2024 paper · derivatives & tangents

Question 1(b) 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.

A curve is defined by the equation y = (x2 + 3x + 2) sin x. Find the gradient of the tangent to this curve when x = 0.

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. 1 Which rule does a bracket times sin x need? Name u and v.

    The product rule: u = x2 + 3x + 2, u' = 2x + 3; v = sin x, v' = cos x.

  2. 2 Write the derivative.

    dy/dx = (2x + 3) sin x + (x2 + 3x + 2) cos x

  3. 3 Put x = 0 in.

    sin 0 = 0 and cos 0 = 1, so dy/dx = 3 × 0 + 2 × 1 = 2.
Answer The tangent’s gradient at x = 0 is 2.

What the marker wanted

  • Achieved A gradient of 2, from a correct derivative.

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

How did it go?

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