3. Take the inverse

5. Example:

8. The derivatives of the remaining trigonometric functions — csc, sec, and cot — can also be found easily using the Quotient Rule. All together:

9. Example: Differentiate For what values of x does the graph of f have a horizontal tangent?

10. Since sec x is never 0, we see that f’(x) = 0 when tan x = 1. This occurs when x = nπ + π/4, where n is an integer

11. Example: Find Example: Calculate:

12. The differentiation formulas you learned by now do not enable you to calculate F’(x). It turns out that the derivative of the composite function f ◦ g is the product of the derivatives of f and g. Proof goes over the head, so forget about that. This fact is one of the most important of the differentiation rules. It is called the Chain Rule.

13. If u changes twice as fast as x and y changes three times as fast as u, it seems reasonable that y changes six times as fast as x. So, we expect that:

14. Chain Rule Chain Rule easy to remember because, if dy/du and du/dx were quotients, then we could cancel du. However, remember: du/dx should not be thought of as an actual quotient

15. Let’s go back: In order not to make your life too complicated (it is already enough), we’ll introduce one way that is most common and anyway, everyone ends up with that one: Leibnitz dy/dx refers to the derivative of y when y is considered as a function of x (called the derivative of y with respect to x) dy/du refers to the derivative of y when considered as a function of u (the derivative of y with respect to u)

16. example:

17. Differentiate y = (x 3 – 1) 100 example: Taking u = x 3 – 1 and y = u 100

18. example: Find f’ (x) if First, rewrite f as f(x) = (x 2 + x + 1) -1/3 Thus,

19. Find the derivative of example: Combining the Power Rule, Chain Rule, and Quotient Rule, we get:

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

21. 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:

22. example: Notice that we used the Chain Rule twice.

23. Differentiate example:

24. 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: