1. The Derivative

2. Recall: The slope of the tangent line to the graph of f(x) at any value of x can be found using limits and the difference quotient: Put another way, we are looking at the slope of the secant line when the two endpoints used in the calculation of slope are an infinitesimal distance apart:

3. The Derivative For a function y = f(x), we define the derivative of f(x), , as follows: Note: This is notation. The derivative is not a fraction, so the expression cannot be reduced to . DUH!

4. Using the difference quotient, the derivative can be expressed as follows: What is the geometric meaning of ?

5. Exercises: 1) Find for the function y = 2x3 - 4. 2) For the function f(x) = 1/x, find at x = 4.

6. f(x) = 2x3 - 4 d f(x) = 6x2 dx

7. 3) Find the point at which = -6

8. g(x) = 6x2 d g(x) = 12x dx

9. As you can see from the exercises, is itself a function of x. Since the derivative can be obtained from y = f(x) (and also to indicate that is a function of x), the derivative is also often denoted by .

10. Exercise: For h(x) = 12x, find h'(-1).

11. h(x) = 12x h'(x) = 12

12. Another way to express the derivative of the function f:

13. Definitions 1) The function y = f(x) is differentiable at a if f'(a) exists. 2) The function y = f(x) is differentiable if f'(x) exists for every real number x.

15. The derivative does not exist!

18. Practice: Explain why the following function is not differentiable everywhere. Use the definition of the derivative to justify your response.

19. One more definition: The normal to the graph of f at point P is the line that is perpendicular to the tangent line at point P.

20. How will the derivative function look?

22. Attachments