Functions of several variables

When a function depends on several variables, a partial derivative measures how it changes as one variable changes while the others are held fixed.

Key facts

  • ∂f/∂x: differentiate with respect to x, treating y as a constant.
  • The gradient vector ∇f = (∂f/∂x, ∂f/∂y) points uphill fastest.
  • Stationary points: both partial derivatives zero.
  • Double integrals find volumes under surfaces.

Worked examples

Example 1

f(x, y) = x²y + 3y². Find ∂f/∂x and ∂f/∂y.

  1. Treat y as a constant: ∂f/∂x = 2xy.
  2. Treat x as a constant: ∂f/∂y = x² + 6y.

Answer: 2xy and x² + 6y