14.4 Chain Rules for functions of several variables

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Presentation transcript:

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

1st type of Chain Rule

Dependent on 2 variables…

Generalization to higher dimensions…

2nd Type of Chain Rule: Implicit differentiation

Old stuff from Calculus I and II

Same problem from before but now using 2nd method…

Harder example

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?