Worked examples: TCE Mathematics Methods Information Sheet

Selected worked examples

1. Calculus · derivative of a logarithm

   Formula or reference: d(ln x)/dx = 1/x for x > 0.

   Units and conditions: When the logarithm has an inner function, apply the chain rule.

   Worked example: For y = ln(4x + 1), y′ = 4/(4x + 1). At x = 1 the gradient is 0.8.

2. Statistics · standardising a score

   Formula or reference: z = (x − μ)/σ

   Units and conditions: σ must be positive. x, μ and σ use the same units.

   Worked example: A score of 74 in a population with mean 62 and standard deviation 8 gives z = 1.5.