The document discusses techniques for taking derivatives of composite functions using the Chain Rule. It explains that to differentiate a composite function f(g(x)), take the derivative of the outside function f evaluated at the inside function g(x), and multiply by the derivative of the inside function g(x). Several examples are provided to illustrate how to identify the inside and outside functions of composites, and apply the Chain Rule along with other derivative rules like the General Power Rule.

1. 2.4 The Chain Rule Derivatives of Composite Functions Using the General Power Rule

2. The key to success is identifying the outside function and the inside function! Take the derivative of the outside function evaluated at the inside function, then chain on the derivative of the inside function with respect to x.

3. Think of the composite function as having two parts – the outside and the inside. y = f( g(x) ) = f(u) and y’ = f’(u) ٠ u’ Ex. 2 p. 132 Decomposition of a Composite Function – identifying inside parts and outside parts!

4. Ex 3 p 132 Using the Chain Rule Find dy/dx for y = (3x 2 + 1) 7 Solution: inside function is u = 3x 2 + 1 and outside is y = u 7 For f(g(x)): Differentiate the “outside” function f and evaluate it at the “inside” function g(x) left alone, then multiply by the derivative of the inside function .

6. Ex 5 p. 133 Differentiating Functions with radicals Find all points on the graph of for which f’(x) = 0 and those for which f’(x) does not exist. Rewrite: Apply chain rule: f’(x) = 0 when the numerator = 0 so when x = 0 f‘(x) does not exist when denominator = 0 so when x = 3, -3

7. Ex 6 p. 133 Differentiating Quotients with Constant Numerators Differentiate Rewrite as Then apply General Power Rule:

8. Ex 7 p134 Simplifying by Factoring Out the Least Powers Way 1 Way 2

9. Ex 8 p. 134 Simplifying the Derivative of a Quotient Way 1 Way 2

10. Example 9 p. 134 Simplifying the Derivative of a Power

11. 2.4a p. 137 #1-3 all, 7-35 EOO, 59-67 EOO, 91-97