The Chain Rule for Partial Derivatives

Return to List of Lectures

Previous Lecture

• Questions?
• The Chain Rule
  • Section 14.4
    • dw/dt = dw/dx dx/dt
    • With partial derivatives: dw/dt = ∂w/∂x dx/dt + ∂w/∂y dy/dt
    • Extends to z for 3 variables
    • Functions on surfaces are almost same but with ∂w/∂r = ∂w/∂x ∂x/∂r + ∂w/∂y ∂y/∂r + ∂w/∂z ∂z/∂r
    • Implicit differentiation
  • Examples
    • Use the chain rule to find dz/dt given z(x,y) = yx² + xy³ and x(t) = e^t, y(t) = ln t
      
    • Find dz/dt above without using the chain rule
      
      • Produces same result as using the chain rule
      • Chain rule not essential for evaluating such things, and isn’t necessarily even easier; its value is often more as a tool for proving other things than for direct calculation
    • Use the chain rule to find ∂f/∂u and ∂f/∂v given f( x, y ) = sin²x + cos²y, x = 2u + v³, y = u² - 3v
      
    • Use the book’s implicit differentiation shortcut to find dy/dx if xy² - yx² = 0
      • f( x, y ) = xy² - yx²
      • dy/dx = - ∂f/∂x / ∂f/∂y
• Next
  • Directional derivatives and gradients
  • Read section 14.5

Next Lecture