1. The Derivative

2. Def: The derivative of a function f at a number a, denoted f’(a) is: Provided this limit exists.

3. If that limit looked familiar, it should! It is the same limit as the one for finding the slope of the tangent line to a function at a point.

4. Some Books: Provided this limit exists.

5. EX: For what function and at what point would this limit represent the derivative?.

7. EX: Suppose the equation of the tangent line to a function f(x) at x=4 is y=2x+3. What is f’(4)? slope

8. EX: Suppose the tangent line to a function f(x) at (3,2) also passes through the point (0,-1) Find: f(3) and f’(3)

9. Alternate form: Provided this limit exists.

10. EX: For what function and at what point would this limit represent the derivative?.

12. EX: For some functions this limit is easier to evaluate: They both will show f’(1)=2

13. The derivative function: Provided this limit exists.

14. The Domain of the Derivative Function:

15. How can a function not have a derivative at a point ? Clearly if the function is not defined at a point then no derivative exists there.

16. The Domain of the Derivative Function: -Would only consist of x values that were also in the Domain of the function (How can there be slope without a curve?)

17. How can a function not have a derivative at a point? There is a vertical tangent line at x=0. The slope and the derivative at x=0 are undefined.

18. The Domain of the Derivative Function: -But may not exist for all of those values. -Would only consist of x values that were also in the Domain of the function (How can there be slope without a curve?)

19. How can a function not have a derivative at a point ? Also if the function is discontinuous at a point then no derivative exists there. Here, there is no derivative at x=0 since it would matter from what side x approaches zero as to what slope you’d get.

20. How can a function not have a derivative at a point? Even though this function is continuous, the derivative at x=0 does not exist. It matters from what side of zero x approaches as to what the slope is. (There is an abrupt change of slope at x=0, not a gradual one.)

21. Notations:

22. Notation:

23. Terminology: T The derivative (n.) Instantaneous rate of change (n.) Slope of the tangent line (n.) Derive (v.) Differentiate (v.) Differentiable (adj.)

24. Old Terminology: T Average rate of change Slope of the secant line

25. Def: A function f is differentiable at a if f’(a) exists. It is differentiable on an open interval if it is differentiable at every number in the interval.

26. Theorem: If f is differentiable at a, then it is continuous at a. Is the converse true?

27. NO! Counterexample: If f is continutous at a, then it is differentiable at a. NO!

28. Examples: In the next screens you will be asked to describe the derivative values (slopes) on parent functions by answering: or =

29. Fill in the blank with: >

30. <

31. >

32. <

33. =

34. =

35. <

36. >

37. >

38. >

39. >

40. <

41. >

42. <

43. Undefined

44. Given the function: Find the points on the graph where there is a horizontal tangent line.

45. Using a Calculator to graph a derivative function: Math (8) Vars Function Y1