Differentiation

Differentiation finds the gradient of a curve at any point: how fast one thing changes compared with another. For powers of x there’s a simple rule.

Key facts

  • If y = xⁿ, then dy/dx = nxⁿ⁻¹.
  • Constants disappear; d/dx(kx) = k.
  • Differentiate each term separately.
  • From first principles: f′(x) = lim_(h→0) (f(x + h) − f(x))/h.
  • Tangent at a point: gradient from dy/dx, then y − y₁ = m(x − x₁).

Worked examples

Example 1

Find the gradient of y = x³ − 4x + 1 at x = 2.

  1. dy/dx = 3x² − 4
  2. At x = 2: 12 − 4 = 8

Answer: 8

Example 2

Differentiate f(x) = x² from first principles.

  1. (f(x + h) − f(x))/h = ((x + h)² − x²)/h
  2. = (2xh + h²)/h = 2x + h
  3. As h → 0: f′(x) = 2x

Answer: 2x