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12.5 Chain Rules for functions of several variables

12.5 Chain Rules for functions of several variables. Use the chain rule for functions of several variables Find partial derivatives implicitly. 1 st type of Chain Rule. Dependent on 2 variables…. Generalization to higher dimensions…. 2 nd Type of Chain Rule: Implicit differentiation.

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12.5 Chain Rules for functions of several variables

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  1. 12.5 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

  3. Dependent on 2 variables…

  4. Generalization to higher dimensions…

  5. 2nd Type of Chain Rule:Implicit differentiation

  6. Old stuff from Calculus I and II

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

  8. Harder example

  9. 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?

  10. Homework Pg. 883 #1-9 odd, 15-19 odd, 23, 27, 29 31, 39, 41

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