The Derivative of a Function Measuring Rates of Change of a function. Secant line. f(x) f(x 0 ) Average rate of change of with respect to over,

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1 The Derivative of a Function Measuring Rates of Change of a function y f(x) f(x 0 ) P Q Secant line x 0 x x Average rate of change of with respect to over, " " " " - Slope of secant line through, and, 71

2 Instantaneous rate of change of with respect to at point Tangent line y f(x) f(x 0 ) P Q Secant line I x 0 x lim x - Slope of tangent line at, [provided the limit exists] 72

3 Slope of Tangent Lines Definition: So, lim lim [provided the limit exists] Since the tangent line passes through,, its equation is Alternate notation: x 0 ; lim 73

4 What is a Derivative Definition: The function [ prime of ] derived from and defined by lim is called the derivative of with respect to (wrt) 74

5 Exercise: Find, if 1 75

6 Solution: lim lim 1 1 lim lim 2 76

7 Exercise: Let check that the tangent slope of is "" everywhere 77

8 Solution: lim lim 1 lim 1 lim 78

9 Functions: Differentiable (or not!) at a single point? We say: is differentiable at [has a derivative at if ) exists. The process of finding derivatives of function is called differentiation If a function has a derivative at a point it is said to be differentiable at that point e.g. is differentiable at every point in its domain except 0 Geometric reason: 79

10 A function differentiable at a point is continuous at that point Theorem: If is differentiable at then is continuous at Proof: Since is differentiable at we know exists lim To show is continuous at we must show [definition of a continuous function] We can rewrite: lim lim 0 80

11 Rewriting once more, we need to show with lim 0 lim lim lim 0 0 So, if is not continuous at, then is not differentiable at 81

12 can fail to be differentiable! Here are the ways in which can fail to be differentiable at Example: is not differentiable at 0 which does not exist because 0 0 lim lim 1, 0 1, 0 82

13 Functions Differentiable on an Interval On open intervals: a function must be differentiable at each point (2-sided limit) On interval with endpoints: a function must be differentiable at each point on the open interval (2-sided limit) and have a left/right hand limits at the end points Definition: Left Hand Derivative Right Hand Derivative lim lim 83

14 Other Derivative Notations If, At So, lim lim 84

15 Finding Derivatives 1. Differentiation technique: lim 2. The derivative of any constant function is zero 0 Obvious: Horizontal line has a horizontal tangent at each point 85

16 3. The Power Rule: For any real number Proof for positive integers, 0, 1,2, Recall: 86

17 lim lim 1 lim 87

18 Examples: function 1. derivative

19 Constance Multiple, Sum and Difference Rules Theorem: If, are differentiable at and is any real number Then 89

20 Exercise: Find 2,

21 Solution: function 2, 1. derivative 0,5, 0,5,

22 The Product Rule Observe: Example: 1, 0, 1,

23 Theorem: If, are differentiable at then 93

24 Proof: lim lim lim lim lim lim lim lim 94

25 Sometimes we write Generalized Product Rule: 95

26 Example: 2 1 Solution: or:

27 The Quotient Rule Observe: Example: 1, 0, 1,

28 Theorem: If, are differentiable at, Then We also write: 98

29 Handy fact:

30 Example: 3 5 Solution

31 The Chain Rule: Derivatives of Composition of functions Motivating example: 1. Find Our only technique is to multiply this out weary tedious. Instead, think of 1 as the composition of two functions. Suppose 1 Then 1 1 We can use the derivatives of and 1 to calculate the derivative of 1 101

32 Rewrite as 1, 1 Then To get we multiply 100,

33 Theorem [The Chain Rule] If is differentiable at and is differentiable at Then is differentiable at 103

34 Exercise: Find 4 1 Solution: ;

35 Derivatives of Trigonometric Functions Recall: Then lim 1; lim 1 0 sin lim sin lim 1 lim 1 lim lim 105

36

37 Exercises: Find function 2 1. derivative 22 sin

38 Derivatives of Inverse Trigonometric Functions

39 Derivatives Involving Logarithms We find For 0 [Domain of ] We need two facts to recall: 1. is continuous So, at any we have: The limit moves through the lim lim 109

40 2. Definition: is that number which 1 approaches, as 2,71828 lim 1 1 Let, so 0 0 Thus [Limit is two-sided] lim 1 110

41 So, ln lim lim 1 ln lim 1 ln 1 Let so 0, 0 lim 1 ln1 1 lim ln1 1 ln lim

42 So, 1, 0 Generalized version: 1, 0 112

43 So, 1, 0 log, 0 log 113

44 Exercises: function derivative ,

45 Derivatives of Exponential Functions What is, 0, 0, 1? 115

46 Development: Since

47 Important case: If 117

48 Exercises: function 1. derivative

49 Notation for Derivatives of Derivatives [Higher order Derivatives] 1 st Derivative: 2 nd Derivative:,,,,,, The second derivative of wrt For higher derivatives, The differentiations rules are the same 119

50 Exercise: ,6,7 120