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
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
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
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
Exercise: Find, if 1 75
Solution: lim lim 1 1 lim 1 2 1 2 1 lim 2 76
Exercise: Let check that the tangent slope of is "" everywhere 77
Solution: lim lim 1 lim 1 lim 78
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
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
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
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
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
Other Derivative Notations If, At So, lim lim 84
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
3. The Power Rule: For any real number Proof for positive integers, 0, 1,2, Recall: 86
lim lim 1 lim 87
Examples: function 1. derivative 2 2 3 3 88
Constance Multiple, Sum and Difference Rules Theorem: If, are differentiable at and is any real number Then 89
Exercise: Find 2, 5 3 6 90
Solution: function 2, 1. derivative 0,5, 0,5, 5 3 10 3 6 6 6 18 9 2 91
The Product Rule Observe: Example: 1, 0, 1, 0 1 0 1 0 92
Theorem: If, are differentiable at then 93
Proof: lim lim lim lim lim lim lim lim 94
Sometimes we write Generalized Product Rule: 95
Example: 2 1 Solution: 3 2 1 2 1 6 6 2 8 6 or: 2 4 2 3 8 6 96
The Quotient Rule Observe: Example: 1, 0, 1, 1 1 0 0 1 0 97
Theorem: If, are differentiable at, Then We also write: 98
Handy fact: 1 0 1 99
Example: 3 5 Solution 3 25 3 1 5 30 6 3 5 3 30 5 100
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
Rewrite as 1, 1 Then To get we multiply 100, 2 100 1 2 200 1 102
Theorem [The Chain Rule] If is differentiable at and is differentiable at Then is differentiable at 103
Exercise: Find 4 1 Solution: 4 1 4 ; 1 4 2 12 16 16 16 1 104
Derivatives of Trigonometric Functions Recall: Then lim 1; lim 1 0 sin lim sin lim 1 lim 1 lim lim 105
1 1 106
Exercises: Find function 2 1. derivative 22 sin 2 2 107
Derivatives of Inverse Trigonometric Functions 1 1 1 1 1 1 1 1 108
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
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
So, ln lim lim 1 ln lim 1 ln 1 Let so 0, 0 lim 1 ln1 1 lim ln1 1 ln lim 1 1 1 111
So, 1, 0 Generalized version: 1, 0 112
So, 1, 0 log, 0 log 113
Exercises: function 2 1 1. derivative 1 2 1 2 1 0,5 1 1 1 3 3 114
Derivatives of Exponential Functions What is, 0, 0, 1? 115
Development: Since 1 1 116
Important case: If 117
Exercises: function 1. derivative 5 5 2 5 2 2 1 2 22 2 322 2 2 322 118
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
Exercise: 3 2 4 2 12 6 2 4 36 12 2 72 12 72 0 5,6,7 120