1. 12.5 Chain Rules for functions of several variables Use the chain rule for functions of several variables Find partial derivatives implicitly

2. 1 ST TYPE OF CHAIN RULE

8. Dependent on 2 variables…

11. Generalization to higher dimensions…

13. 2 ND TYPE OF CHAIN RULE: IMPLICIT DIFFERENTIATION

14. Old stuff from Calculus I and II

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

19. Harder example

20. 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?

21. PG. 883 #1-9 ODD, 15-19 ODD, 23, 27, 29 31, 39, 41 Homework