Chain, product and quotient rules

To differentiate more complicated functions you need three rules: the chain rule for a function inside a function, the product rule for two functions multiplied, and the quotient rule for one divided by another.

Key facts

  • Chain: dy/dx = dy/du × du/dx.
  • Product: (uv)′ = u′v + uv′.
  • Quotient: (u/v)′ = (u′v − uv′)/v².
  • d/dx eˣ = eˣ, d/dx ln x = 1/x, d/dx sin x = cos x, d/dx cos x = −sin x (x in radians).

Worked examples

Example 1

Differentiate y = (3x + 1)⁵.

  1. Let u = 3x + 1, so y = u⁵.
  2. dy/du = 5u⁴, du/dx = 3
  3. dy/dx = 15(3x + 1)⁴

Answer: 15(3x + 1)⁴

Example 2

Differentiate y = x² sin x.

  1. u = x², v = sin x.
  2. u′ = 2x, v′ = cos x
  3. dy/dx = 2x sin x + x² cos x

Answer: 2x sin x + x² cos x