Section 2.8 The Derivative as a Function

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

2. Introduction • 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 avary:

3. Introduction (cont’d) • 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 . • Note that f ´(x) is a limit. • Thus f ´(x) is defined only when this limit exists.

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

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

6. Solution (cont’d)

7. Solution (cont’d)

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

9. Example • For the function f(x) = x3 – x , • Find a formula for f ´(x) • Compare the graphs of f and f ´ • Solution On the… • next slide, we show that f ´(x) = 3x2 – 1 ; • following slide, we give the graphs of f and f ´ side-by-side:

10. Solution (cont’d)

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

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

13. Solution (cont’d)

14. Other Notations • 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.

15. Other Notations (cont’d) • 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: • Also we can indicate the value f ´(a) of a derivative dy/dx as

16. Differentiability • We begin with this definition: • 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? • Solution If x > 0 , then… • |x| = x and we can choose h small enough that x + h > 0 , so that |x + h| = x + h • Therefore

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

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

20. Solution (cont’d) • We can give a formula for f ´(x) : • 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 . • However, as our preceding example shows, the converse is false: • The function f(x) = |x| • is continuous everywhere, but • is not differentiable at x = 0 .

23. Failure of Differentiability • 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… • …as does the graph of f(x) = |x| ; • f can be discontinuous 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 . • 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. • We zoom in toward the point (a, f(a)) : • If fis differentiable at x = a , then the graph • straightens out and • appears more and more like a line. • 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… • 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 . • In Leibniz notation the second derivative of y = f(x) is written

31. Example • If f(x) = x3 – x , find and interpret f (x) . • Solution We found earlier that the first derivative f ´(x) = 3x2 – 1 . • On the next slide we use the limit definition of the derivative to show that f (x) = 6x :

32. Solution (cont’d)

33. Solution (cont’d) • 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)) . • That is, f (x) is the rate of change of the slope of the original curve y = f(x) . • 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 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… • its first derivative gives the velocityv(t) of the object: • The accelerationa(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. • Here s is measured in feet and t in seconds. • Use this to graph the velocity and acceleration of the car. • What is the acceleration at t = 2 seconds?

37. Position Function of a Car

38. Solution • By measuring the slope of the graph ofs = 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 slope of the tangent line to the graph of f ´ when t = 2 . • The slope of this tangent line is

39. Velocity Function

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

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

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

43. Example • If f(x) = x3 – 6x , find f (x) and f(4)(x) . • Solution Earlier we found that f (x) = 6x . • The graph of y = 6x is a line with slope 6 ; • 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 , f(4)(x) = 0

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