1. 14.4 Chain Rules for functions of several variables
Use the chain rule for functions of several variables
Find partial derivatives implicitly

2. 1st type of Chain Rule

8. Dependent on 2 variables…

11. Generalization to higher dimensions…

12. 2nd Type of Chain Rule: Implicit differentiation

13. Old stuff from Calculus I and II

15. Same problem from before but now using 2nd method…

18. Harder example

19. Applications
The portion of a tree usable for lumber may be viewed as a right circular cylinder. If the tree’s height increases as a rate of 24 inches/year and the radius increases at a rate of 3 inches/year, at what rate is the tree’s volume growing when it has a height of 240 inches and a radius of 30 inches?