15.4 The Chain Rule

$\frac{dz}{dt} = \frac{\delta z}{\delta x} \frac{dx}{dt}+ \frac{\delta z}{\delta y} \frac{dy}{dt}$

• With two independent variables
  • $\frac{dz}{ds} = \frac{\delta z}{\delta x} \frac{dx}{ds}+ \frac{\delta z}{\delta y} \frac{dy}{ds}$
  • $\frac{dz}{dt} = \frac{\delta z}{\delta x} \frac{dx}{dt}+ \frac{\delta z}{\delta y} \frac{dy}{dt}$
• Implicit Diffrentiation: $\frac{dy}{dx} = - \frac{F_x}{F_y}$

15.5 Directional Derivatives and the Gradient

Assigned HW: 3,4,5,15,19,23,29,31,35,41,45,47,53,61,63,67,81,82

Directional Derivative

Directional derivative

Gradient

Gradient

Gradient and Level Curves

The Gradient in Three Dimensions

15.6 Linear Approximation