1. Chapter 16 Section 16.3 The Mean-Value Theorem; The Chain Rule

2. Chain Rule for a 1 Variable Function The derivative of a complicated function that can be formed by substitutions (compositions) of simpler functions can be found by apply the chain rule. That is taking the derivatives of each simpler function and multiplying them together. Dependence tree y u v x y depends on u u depends on v v depends on x Dependence tree z xy s s t t

3. Example The variables u, v, w, x, y, and z all depend on each other as given to the right. Draw the dependence tree and give the partial derivatives using partial derivative and using the variables. Dependence tree w xy z uv uv v With the variables : Notice each position in the tree that ends with the independent variable that you are taking the derivative with respect to represents a term in the partial derivative. Each tier (level) of the tree represents a factor in that term. Dependence tree z xy t t

4. w x y z u v r u r