Worked examples: QCE Mathematical Methods Formula Book

Selected worked examples

1. Calculus · derivative of a product

   Formula or reference: (uv)′ = u′v + uv′

   Units and conditions: Both factors are differentiable functions of the same variable.

   Worked example: If y = xeˣ, then y′ = eˣ + xeˣ, so the gradient at x = 0 is 1.

2. Probability · binomial distribution

   Formula or reference: P(X = k) = C(n,k)pᵏ(1 − p)ⁿ⁻ᵏ

   Units and conditions: Use independent trials with a common success probability.

   Worked example: Exactly two successes in four trials at p = 0.25 has probability 6 × 0.25² × 0.75² ≈ 0.211.