Chapter 12: Differentiation. SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M.

Transcription

1 Chapter 12: Differentiation SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza

2 Chapter 12: Differentiation Lecture 12.1: The Derivative Lecture 12.2: Differentiation Rules Lecture 12.3: The Chain Rule

3 Chapter 12: Differentiation Lecture 12.4: The Power of a Function Rule Lecture 12.5: The Higher Order Derivatives Lecture 12.6: Derivatives of Exponential Function

4 Chapter 12: Differentiation Lecture 12.7: Derivatives of Logarithmic Function Lecture 12.8: Derivatives of Trigonometric Functions

5 Lecture 12.1: The Derivative SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza

6 Recall: What is the slope of a line? What does the slope of a line tell us about the direction of the line?

7 Recall: What is the formula for finding the slope of a line?

8 Understanding Derivative: Consider a function y f (x) defined in an open interval and a point a in the interval shown in the figure on the next slide. Let be a point Pa, f ( a y f (x) Qx, f ( x) on the curve and be another point on the curve. The line that is drawn through these two points is called a secant line. )

9 Figure 12.1:

10 Slope of the Secant Line The slope of the secant line through P and Q is given by: f ( x) f ( a) m x a

11 Understanding Derivative: Now, allow the second point Q to approach the first point P. The secant lines approach the line t. Notice that the line t touches the graph of f at exactly one point P. A line that touches the function at exactly one point is called a tangent line.

12 Figure 12.2:

13 Figure 12.3:

14 Slope of the Tangent Line To find the slope of the tangent line t to the curve y f (x), we want the value x to approach the value a, as this will cause the point x, f ( x) to approach the point a, f ( a). So the slope of the tangent line to the curve y f (x) through P will be: m lim xa f ( x) x provided that this limit exists. f a ( a)

15 Slope of the Tangent Line Now, if we let the distance between a and x be represented by h, we have x a h. This gives us the following definition of the slope of the tangent line: f ( a h) f ( a) m lim h0 h

16 The Slope of a Tangent Line If the point P a, f ( a) is on the curve of the function f, then the slope of the tangent line m through P is given by: lim h0 f ( a h) h ( a) provided that this limit exists. f

17 Example : Find the equation of the line tangent to the graph of f ( x) 2x 2 3x 4 at the point where x = 2.

18 Recall: What are the different types of equation of a line we have discussed in Precalculus?

19 Final Answer: Thus, from the point-slope form, we have and therefore the equation of the tangent line is y 6 5( x y 2), 5x 4. The graph is shown on the next slide.

20 Figure 12.4:

21 Take Note: f The slope of the tangent line at a point a on the function is equal to the derivative of the function at the same point, denoted by f (a), read as f prime of a. Hence, from our previous example, '(2) 6. If we let a be arbitrary and assume a general value in the domain of f, the derivative f is a function.

22 Take Note: Obviously, the slope is different at different points on the curve. Thus, the derivative at any point x on the curve f(x) is given by:

23 The Derivative of a Function The derivative of f(x) with respect to x is the f '( x) function f (x) where: lim h0 f ( x h) h ( x) provided this limit exists. The domain of f consists of all x in the domain of f for which the limit exists. f

24 Did you know? The idea of the slope of the tangent line to the graph of a function at a point is the geometric method of explaining the meaning of the derivative of a function.

25 Did you know? Every time you get the derivative of a function, you are actually determining the slope of the tangent line at a point x = a.

26 Example : Find the derivative f (x) of the function f ( x) x, x 0.

27 Final Answer: Note that f ( x) x is defined for all x 0, whereas its derivative f (x) is defined only x > 0. From this, we can see that a function need not have a derivative throughout its entire domain.

28 Example : Find the derivative of f ( x) 4 x 2 x.

29 Final Answer: Therefore, the derivative of f is f ' ( x) 8x 1.

30 Example : (a) (b) (c) Find f (x); f Given ( x) find an equation of the tangent line to the graph of f(x) at the point where x = 2; and find an equation of the line that is perpendicular to the tangent line to the graph of f(x) at x = 2 and intersects it at the point of tangency. 1 x,

31 Final Answer for (a): Using the definition, 1 f '( x) x 2

32 Final Answer for (b): The tangent line to the graph of f(x) is y 1 x 1. 4

33 Slope Property Number 4: Distinct lines of the same slope are parallel. Conversely, if two nonvertical lines are parallel, they have the same slope.

34 Slope Property Number 4: Suppose two nonvertical lines l 1 and l 2 are perpendicular, having respective slopes m 1 and m 2, the slope of one of the lines is the negative reciprocal of the slope of other line: m 2 1 m 1

35 Final Answer for (c): Thus, we have 1 y 4( x 2 2) and the required equation is 15 y 4x. 2

36 Figure 12.5:

37 Normal Line The line in the previous example is called a normal line. The normal line to the graph of f at point P is the line that is perpendicular to the tangent at P. Thus, it intersects the tangent at the point of tangency.

38 Differentiability A function f is said to be differentiable at a if f (a) exists. At points where f is not differentiable, we say that the derivative does not exist.

39 Take Note: The derivative does not exist at any point where the function is not continuous. In other words, wherever the function is not continuous, it is also not differentiable.

40 Did you know? But even if the function is continuous at a point, it still might not be differentiable there.

41 Did you know? There are three conditions which must be fulfilled for the function to be differentiable at a given point.

42 First Condition: The function must be continuous there.

43 Figure 12.5: Discontinuity

44 Second Condition: The graph must be smooth there, with no sharp bend.

45 Figure 12.6: Cusp

46 Third Condition: The tangent line must be oblique (slanted) or horizontal, not vertical.

47 Figure 12.7: Vertical Tangent

48 Take Note: Whenever any of those conditions is violated, the derivative does not exist at that point. However, if the derivative exist at a given point, then it follows that all three of the above are true at that point. Hence, if the function is differentiable at a point, it is continuous at that point.

49 Example Show that the absolute value f ( x) x differentiable at x = 0. function is not

50 Figure 12.8.

51 Final Answer: lim h0 h h Using one-sided limits, we have lim h0 h h 1 and. lim h0 Since the left-hand and right-hand limits are not the same, the derivative at x = 0 does not exist. h h lim h0 h h 1

52 Something to think about What have you observed from the previous example?

53 Our Observation: From the previous example, we conclude that it is possible for a function to be continuous at a point and yet not differentiable at that point.

54 Did you know? Symbols other than f (x) are often used to denote the derivative. y f (x), dy If y the symbols and. f (x). are used instead of dx

55 How to read? y y prime dy the derivative of y dx with respect to x

56 Take Note: dy It is important to note that is dx not a fraction. It is a notation which represents the derivative of y with respect to x.

57 Classroom Task 12.1: Please answer "Let's Practice (LP)" Numbers 45.

58 Lecture 12.2: Differentiation Rules SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza

59 Did you know? Determining derivatives based on the definition, as we did in Lecture 12.1, is tedious and time-consuming. In this lecture, we will discuss some rules that simply the process of differentiation.

60 Classroom Task: Using the definition of the derivative, derive the differentiation rules for constant function, where k is any real number: f ( x) k

61 The Constant Function Rule If f ( x) k, where k is a constant, then f ' ( x) 0.

62 Example : Given, f ( x) find f (x). 2

63 Final Answer: Using constant function rule, we have: f ' ( x) 0.

64 Example : Given, f ( x) 2 find f (x).

65 Final Answer: Thus, f ' ( x) 0.

66 Homework: Using the definition of the derivative and the factors of the difference of two nth powers where n is a positive integer, derive the differentiation rules for power function: n f ( x) x.

67 The Power Rule If f ( x) x n, where n is a real number, f '( x) nx n1 then..

68 Example : Find the derivative f (x) of the given functions: ( a) f ( x) x 9

69 Final Answer: With n = 9, we have: f '( x) 9 x 8.

70 Example : Find the derivative f (x) of the given functions: 3 2 ( b) f ( x) x

71 Final Answer: Rewriting the given function, we have: 2 f '( x). 3 3 x

72 Example : Find the derivative f (x) of the ( c) given functions: f ( x) 1 x 3

73 Final Answer: Using power rule, we have: f 3 '( x) x 4.

74 Example : Find the derivative f (x) of the given functions: ( d ) f ( x) x

75 Final Answer: Using power rule, we have: f ' ( x) 1.

76 Classroom Task: Using the definition of the derivative, derive the differentiation rules for constant multiple function, where k is a constant: f ( x) kg( x)

77 The Constant Multiple Rule If f ( x) kg( x), where k is a constant, then f ' ( x). kg'( x).

78 Example : Differentiate the following functions: ( a) f ( x) 5x 4

79 Final Answer: Using constant multiple rule, now we have: f '( x) 20 x 3.

80 Example : Differentiate the following functions: ( b) y 8x 3 5

81 Final Answer: By applying constant multiple rule, we have: 24 x f '( x)

82 Example : Differentiate the following functions: ( c) y 4 x

83 Final Answer: Rewriting the given function, we have: f 4 '( x) x 2.

84 Classroom Task: Suppose we are given two differentiable functions p(x) and q(x), Using the definition of the derivative, derive the differentiation rules for finding their sum: f ( x) p( x) q( x)

85 The Sum Rule If the functions p(x) and q(x) are differentiable and f ( x) p( x) q( x), then: f ' ( x) p'( x) q'( x).

86 Classroom Task: Suppose we are given two differentiable functions p(x) and q(x), Using the definition of the derivative, derive the differentiation rules for finding their difference: f ( x) p( x) q( x)

87 The Difference Rule If the functions p(x) and q(x) are differentiable and f ( x) p( x) q( x), then: f ' ( x) p'( x) q'( x).

88 Example : Differentiate the following ( a) f ( functions: x) 4x 3 5 x 2

89 Final Answer: f Thus, 10 '( x) 12x 2 x 3.

90 Example : Differentiate the following functions: 2x 3 2 ( b) y

91 Final Answer: By expansion, we have, f ' ( x) 8x 12.

92 Classroom Task: Suppose we are given two differentiable functions f(x) and g(x), Using the definition of the derivative, derive the differentiation rule for finding their product: h( x) f ( x) g( x)

93 Take Note: Unlike the Sum and Difference Rule, the derivative of a product of two functions is not the product of the separate derivatives.

94 The Product Rule h( x) f ( x) g( x) If, then: h' ( x) f '( x) g( x) f ( x) g'( x).

95 Take Note: The Product Rule states that the derivative of the product of two functions is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function.

96 Example : Differentiate h( x) ( x 2 5x)( x 5 2), using the Product Rule.

97 Final Answer: Using the Product Rule, we h'( x) 7x 6 get 30 x 5 4x 10

98 Example : Determine the derivative of h( x) (3x 3 2x 2 x 3)( x 4 3x 4)

99 Final Answer: Applying the Product Rule of differentiation: ) 3 ( 3 4 3) 2 (3 ) ( x x x x x x x x x h

100 Classroom Task: From the Product Rule, derive the Quotient f ( x) Rule. Given h( x). g( x) where, we rewrite this as a g( x) 0 product, h( x) g( x) f ( x)

101 The Quotient Rule If where, then: ) ( ) ( ) ( x g x f x h. ) ( ) '( ) ( ) ( ) '( ) ( ' 2 x g x g x f x g x f x h 0 ) ( x g

102 Example : Find the derivative of: 4x h( x) x

103 Final Answer: Applying Quotient Rule, we have: 2 2x 2 3x 4 h'( x) x 2 2 2

104 Classroom Task 12.2: Please answer "Let's Practice (LP)" Numbers 46.

105 Lecture 12.3: The Chain Rule SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza

106 The Chain Rule If f and g are differentiable functions, then the composite function h( x) f g( x) derivative given by: has a h' ( x) f ' g( x) g'( x).

107 Example : Differentiate: x 3 4x. h( x) 2 3

108 Final Answer: By the Chain Rule, we have: h'( x) x 3 4x 3x 2 4 2

109 Example : 2 If y u 3u 2, where u x, find dy dx.

110 Final Answer: By the Chain Rule, we have: dy dx 1 2 x 3 2 x

111 Example : 2 If y 2x 3, find dy dx.

112 Final Answer: By the Chain Rule, we have: dy dx 8x 12

113 Lecture 12.4: The Power Rule SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza

114 Did you know? We have a special case of the chain rule where the outer function is a power function of the form. This is called the Power of a Function Rule. y g( x) n

115 The Power of a Function Rule If n is a real number and y g( x) n, then dy dx n g( x) n1 g'( x).

116 Example : Differentiate. y 2 3 x 1 Express your answer in a simplified factored form.

117 Final Answer: Using the Power of a Function Rule, we have: dy dx 3 x

118 Classroom Task 12.3: Please answer "Let's Practice (LP)" Numbers 47.

119 Lecture 12.5: The Higher Order Derivatives SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza

120 The Higher Order Derivatives Given a differentiable function y f (x), its derivative dy f '( x) dx called the. is first derivative.

121 The Higher Order Derivatives The second derivative is defined to be the derivative of the first derivative and is written as f ''( x) or d 2 dx y 2.

122 The Higher Order Derivatives The third derivative is defined to be the derivative of the second derivative and is written as f '''( x) or d 3 dx y 3.

123 The Higher Order Derivatives In general, the nth derivative is defined to be the derivative of the ( n 1)th derivative.

124 Example : Find d 2 dx y 2 if y 6 x 5 x 3.

125 Example : Find f ''''(. x) given: f ( x) 1 x.

126 Classroom Task 12.5: Please answer "Let's Practice (LP)" Numbers 48.