1. ● one of the most important of the differentiation rules

2. Proof of the Chain Rule

3. Let’s go back: In order not to make your life too complicated (it is already enough), we’ll use one way that is most common and anyway, everyone ends up with that one: Leibnitz form

4. Example 1: I DO Differentiate:

5. a. Differentiate y = (x 3 – 1) 100 Example 2: u = x 3 – 1 and y = u 100 WE DO

6. Combining the Power Rule, Chain Rule, and Quotient Rule, we get: YOU DO

7. Differentiate: y = (2x + 1) 5 (x 3 – x + 1) 4 2. In this example, we must use the Product Rule before using the Chain Rule.

8. The reason for the name ‘Chain Rule’ becomes clear when we make a longer chain by adding another link. Suppose that y = f(u), u = g(x), and x = h(t), where f, g, and h are differentiable functions, then, to compute the derivative of y with respect to t, we use the Chain Rule twice:

9. example 1: Notice that we used the Chain Rule twice. first time second time finish it Differentiate

10. first time second time finish it YOU DO

11. The chain rule enables us to find the slope of parametrically defined curves x = x(t) and y = y(t): Divide both sides by The slope of a parametrized curve is given by: