1. The Chain Rule

2. The Chain Rule Case I z x y t t start with z z is a function of x and y x and y are functions of t Put the appropriate derivatives along the branches Tree diagram Multiply along each path and sum the contributions of both paths.

3. Chain Rule - Example 1

4. Chain Rule - Example 2

5. The Chain Rule Case II z x y tt Put the appropriate derivatives along the branches Tree diagram s s

6. Chain Rule - Example 3 s s t t r z

7. Chain Rule - More general case We can generalize the chain rule to any number of variables. s st t x r r w y st r

8. Chain Rule - Implicit Differentiation 2D Chain rule:

9. Chain Rule - Example 5 (b) Find the equation of the tangent line to the curve at the point (1,1).

10. Chain Rule - Implicit Differentiation 3D

11. Chain Rule - Example 6