1. Section 2.8 The Derivative as a Function Goals Goals View the derivative f ´(x) as a function of x View the derivative f ´(x) as a function of x Study graphs of f ´(x) and f(x) together Study graphs of f ´(x) and f(x) together Study differentiability and continuity Study differentiability and continuity Introduce higher-order derivatives Introduce higher-order derivatives

2. Introduction So far we have considered the derivative of a function f at a fixed number a : So far we have considered the derivative of a function f at a fixed number a : Now we change our point of view and let the number a vary: Now we change our point of view and let the number a vary:

3. Introduction (cont’d) Thus f ´(x) becomes its own, new, function of x, called the derivative of f. Thus f ´(x) becomes its own, new, function of x, called the derivative of f. This name reflects the fact that f ´ has been “derived” from f. This name reflects the fact that f ´ has been “derived” from f. Note that f ´(x) is a limit. Note that f ´(x) is a limit. Thus f ´(x) is defined only when this limit exists. Thus f ´(x) is defined only when this limit exists.

4. Example At right is the graph of a function f. At right is the graph of a function f. We want to use this graph to sketch the graph of the derivative f ´(x). We want to use this graph to sketch the graph of the derivative f ´(x).

5. Solution f ´(x) We can estimate f ´(x) at any x by drawing the tangent at the point (x, f(x)) and estimating its slope. f ´(5) ≈ 1.5. Thus, for x = 5 we draw the tangent at P in Fig. 2(a) (on the next slide), and estimate f ´(5) ≈ 1.5. P ´(5, 1.5)f ´ Then we plot P ´(5, 1.5) on the graph of f ´. Repeating gives the graph in Fig. 2(b).

6. Solution (cont’d)

8. Remarks on the Solution The tangents at A, B, and C are horizontal, so the derivative is 0 there, and f ´A ´ ´ C´ the graph of f ´ crosses the x-axis at A ´, B ´, and C´, directly beneath A, B, and C. Between… f ´(x) A and B, f ´(x) is positive; f ´(x) B and C, f ´(x) is negative.

9. Example For the function f(x) = x 3 – x, For the function f(x) = x 3 – x, Find a formula for f ´(x) Find a formula for f ´(x) Compare the graphs of f and f ´ Compare the graphs of f and f ´ Solution On the… Solution On the… a) next slide, we show that f ´(x) = 3x 2 – 1 ; b) following slide, we give the graphs of f and f ´ side-by-side:

10. Solution (cont’d)

11. Notice that f ´(x) is… Notice that f ´(x) is… zero when f has horizontal tangents, and zero when f has horizontal tangents, and positive when the tangents have positive slope: positive when the tangents have positive slope:

12. Example Find f ´(x) if Find f ´(x) if Solution We use the definition as follows: Solution We use the definition as follows:

13. Solution (cont’d)

14. Other Notations Here are common alternative notations for the derivative: Here are common alternative notations for the derivative: The symbols D and d/dx are called differentiation operators because they indicate the operation of differentiation, the process of calculating a derivative. The symbols D and d/dx are called differentiation operators because they indicate the operation of differentiation, the process of calculating a derivative.

15. Other Notations (cont’d) The Leibniz symbol dy/dx is not an actual ratio, but rather a synonym for f ´(x). The Leibniz symbol dy/dx is not an actual ratio, but rather a synonym for f ´(x). We can write the definition of derivative as: We can write the definition of derivative as: Also we can indicate the value f ´(a) of a derivative dy/dx as Also we can indicate the value f ´(a) of a derivative dy/dx as

16. Differentiability We begin with this definition: We begin with this definition: This definition captures the fact that some functions have derivatives only at some values of x, not all. This definition captures the fact that some functions have derivatives only at some values of x, not all.

17. Example Where is the function f(x) = | x | differentiable? Where is the function f(x) = | x | differentiable? Solution If x > 0, then… Solution If x > 0, then… | x | = x and we can choose h small enough that x + h > 0, so that | x + h | = x + h | x | = x and we can choose h small enough that x + h > 0, so that | x + h | = x + h Therefore Therefore

18. Solution (cont’d) This means that f is differentiable for any x > 0. This means that f is differentiable for any x > 0. A similar argument shows that f is differentiable for any x < 0, as well. A similar argument shows that f is differentiable for any x < 0, as well. However for x = 0 we have to consider However for x = 0 we have to consider

19. Solution (cont’d) We compute the left and right limits separately: We compute the left and right limits separately: Since these differ, f ´(0) does not exist. Since these differ, f ´(0) does not exist. Thus f is differentiable at all x ≠ 0. Thus f is differentiable at all x ≠ 0.

20. Solution (cont’d) We can give a formula for f ´(x) : We can give a formula for f ´(x) : Also, on the next slide we graph f and f ´ side-by-side: Also, on the next slide we graph f and f ´ side-by-side:

21. Solution (cont’d)

22. Differentiability and Continuity We can show that if f is differentiable at a, then f is continuous at a. We can show that if f is differentiable at a, then f is continuous at a. However, as our preceding example shows, the converse is false: However, as our preceding example shows, the converse is false: The function f(x) = | x | The function f(x) = | x | is continuous everywhere, but is continuous everywhere, but is not differentiable at x = 0. is not differentiable at x = 0.

23. Failure of Differentiability A function can fail to be differentiable at x = a in three different ways: A function can fail to be differentiable at x = a in three different ways: The graph of f can have a corner at x = a… The graph of f can have a corner at x = a… …as does the graph of f(x) = | x | ; …as does the graph of f(x) = | x | ; f can be discontinuous at x = a ; f can be discontinuous at x = a ; The graph of f can have a vertical tangent line at x = a. The graph of f can have a vertical tangent line at x = a. This means that f is continuous at a but | f ´ ( x) | has an infinite limit as x  a. This means that f is continuous at a but | f ´ ( x) | has an infinite limit as x  a. We illustrate each of these possibilities: We illustrate each of these possibilities:

24. Corner at x = a

25. Discontinuity at x = a

26. Vertical Tangent at x = a

27. More on Differentiability The next slides illustrate another way of looking at differentiability. The next slides illustrate another way of looking at differentiability. We zoom in toward the point (a, f(a)) : We zoom in toward the point (a, f(a)) : If f is differentiable at x = a, then the graph If f is differentiable at x = a, then the graph straightens out and straightens out and appears more and more like a line. appears more and more like a line. If f is not differentiable at x = a, then no amount of zooming makes the graph linear. If f is not differentiable at x = a, then no amount of zooming makes the graph linear.

28. f Is Differentiable At a

29. f Is Not Differentiable At a

30. The Second Derivative If f is a differentiable function, then… If f is a differentiable function, then… its derivative f ´ is also a function, so its derivative f ´ is also a function, so f ´ may have a derivative of its own, denoted by (f ´)´ = f , and called the second derivative of f. f ´ may have a derivative of its own, denoted by (f ´)´ = f , and called the second derivative of f. In Leibniz notation the second derivative of y = f(x) is written In Leibniz notation the second derivative of y = f(x) is written

31. Example If f(x) = x 3 – x, find and interpret f  (x). If f(x) = x 3 – x, find and interpret f  (x). Solution We found earlier that the first derivative Solution We found earlier that the first derivative f ´(x) = 3x 2 – 1. On the next slide we use the limit definition of the derivative to show that On the next slide we use the limit definition of the derivative to show that f  (x) = 6x :

32. Solution (cont’d)

33. On the next slide are the graphs of f, f ´, and f . On the next slide are the graphs of f, f ´, and f . We can interpret f  (x) as the slope of the curve y = f ´(x) at the point (x, f ´(x)). We can interpret f  (x) as the slope of the curve y = f ´(x) at the point (x, f ´(x)). That is, f  (x) is the rate of change of the slope of the original curve y = f(x). That is, f  (x) is the rate of change of the slope of the original curve y = f(x). Notice in Fig. 11 that Notice in Fig. 11 that f  (x) < 0 when y = f ´(x) has a negative slope; f  (x) < 0 when y = f ´(x) has a negative slope; f  (x) > 0 when y = f ´(x) has a positive slope. f  (x) > 0 when y = f ´(x) has a positive slope.

34. Solution (cont’d)

35. Acceleration If s = s(t) is the position function of a object moving in a straight line, then… If s = s(t) is the position function of a object moving in a straight line, then… its first derivative gives the velocity v(t) of the object: its first derivative gives the velocity v(t) of the object: The acceleration a(t) of the object is the derivative of the velocity function, that is, the second derivative of the position function: The acceleration a(t) of the object is the derivative of the velocity function, that is, the second derivative of the position function:

36. Example A car starts from rest and the graph of its position function in shown on the next slide. A car starts from rest and the graph of its position function in shown on the next slide. Here s is measured in feet and t in seconds. Here s is measured in feet and t in seconds. Use this to graph the velocity and acceleration of the car. Use this to graph the velocity and acceleration of the car. What is the acceleration at t = 2 seconds? What is the acceleration at t = 2 seconds?

37. Position Function of a Car

38. Solution By measuring the slope of the graph of s = f(t) at t = 0, 1, 2, 3, 4, and 5, we plot the velocity function v = f ´(t) (next slide). By measuring the slope of the graph of s = f(t) at t = 0, 1, 2, 3, 4, and 5, we plot the velocity function v = f ´(t) (next slide). The acceleration when t = 2 is a = f  (2)… The acceleration when t = 2 is a = f  (2)… …the slope of the tangent line to the graph of f ´ when t = 2. …the slope of the tangent line to the graph of f ´ when t = 2. The slope of this tangent line is The slope of this tangent line is

39. Velocity Function

40. Acceleration Function In a similar way we can graph a(t) : In a similar way we can graph a(t) :

41. Third Derivative The third derivative f  is the derivative of the second derivative: f  = (f  ). The third derivative f  is the derivative of the second derivative: f  = (f  ). If y = f(x), then alternative notations for the third derivative are If y = f(x), then alternative notations for the third derivative are

42. Higher-Order Derivatives The process can be continued: The process can be continued: The fourth derivative f  is usually denoted by f (4). The fourth derivative f  is usually denoted by f (4). In general, the nth derivative of f is… In general, the nth derivative of f is… denoted by f (n) and denoted by f (n) and obtained from f by differentiating n times. obtained from f by differentiating n times. If y = f(x), then we write If y = f(x), then we write

43. Example If f(x) = x 3 – 6x, find f  (x) and f (4) (x). If f(x) = x 3 – 6x, find f  (x) and f (4) (x). Solution Earlier we found that f  (x) = 6x. Solution Earlier we found that f  (x) = 6x. The graph of y = 6x is a line with slope 6 ; The graph of y = 6x is a line with slope 6 ; Since the derivative f  (x) is the slope of f  (x), we have Since the derivative f  (x) is the slope of f  (x), we have f  (x) = 6 for all values of x. Therefore, for all values of x, Therefore, for all values of x, f (4) (x) = 0

44. Review The derivative as a function The derivative as a function The graph of f derived from the graph of f The graph of f derived from the graph of f Finding formulas for f ´(x) Finding formulas for f ´(x) Differentiability Differentiability Definition Definition Differentiability implies continuity… Differentiability implies continuity… …but not conversely …but not conversely Higher-order derivatives Higher-order derivatives Notation Notation