Worked examples: WACE Mathematics Methods Formula Sheet

Selected worked examples

1. Calculus · chain rule

   Formula or reference: Differentiate the outer function at the inner value, then multiply by the derivative of the inner function.

   Units and conditions: Both functions must be differentiable at the relevant inputs.

   Worked example: For y = (2x + 1)⁴, y′ = 8(2x + 1)³. At x = 1 the gradient is 216.

2. Probability · binomial mean and variance

   Formula or reference: E(X) = np and Var(X) = np(1 − p).

   Units and conditions: n independent trials share the same probability p.

   Worked example: For five trials with p = 0.30, the mean is 1.5 and the variance is 1.05.