CALCULUS CHAPTER 3 DERIVATIVES SECTION 3-1 DERIVATIVES Definition of a Derivative: Different notations: Meaning of the symbols: f(3) vs f ‘(3) vs f” (3) Graphs of f vs f ‘: How to take Derivatives: Power Rule:
CALCULUS CHAPTER 3 DERIVATIVES
SECTION 3-1 DERIVATIVES
Definition of a Derivative:
Different notations:
Meaning of the symbols: f(3) vs f ‘(3) vs f” (3)
Graphs of f vs f ‘:
How to take Derivatives:
Power Rule:
Taking Derivatives: Find 𝒅𝒚
𝒅𝒙 of:
𝒚 = 𝟒𝒙𝟑 − 𝟐𝒙𝟐 − 𝒙 + 𝟏 𝒚 = 𝟑
𝒙𝟒 𝒚 = √𝒙 − 𝟕 𝒇(𝒙) = √𝒙𝟑𝟓
Find the equation of the tangent line of the equation 𝒚 = 𝟑𝒙𝟐 − 𝒙 − 𝟑 at x = 1.
Find the points of the horizontal tangents of the equation: 𝒚 = 𝟑𝒙𝟑 − 𝟗𝒙𝟐.
ASSIGNMENT: Page 101 – 102 #1, 4-12, 15-17
CALCULUS CHAPTER 3 DERIVATIVES
SECTION 3-2 Differentiability
Differentiability:
Where derivatives fail to exist:
A. Corners: 𝒚 = |𝒙 − 𝟐| + 𝟏 B. Cusps: 𝒚 = 𝒙𝟐𝟑
C. Vertical Tangent: 𝒚 = (𝒙 + 𝟏)𝟏
𝟑⁄ D. Jump Discontinuity:
Comparing Right Hand Derivatives with Left Hand Derivatives:
A derivative exists at a value if:
𝒚 = {𝒙𝟐 𝒙 < 1
𝟑𝒙 − 𝟐 𝒙 ≥ 𝟏
Is this function continuous at x = 1?
Is this function differentiable at x = 1?
Differentiability implies Continuity:
If f has a derivative at x = a, then f is continuous at x = a. (But not necessarily the other way around.)
ASSIGNMENT: Page 111 #1-10
CALCULUS CHAPTER 3 DERIVATIVES
SECTION 3-3 Product and Quotient Rule (Day 1)
The Product Rule:
𝒅
𝒅𝒙(𝒖 ∙ 𝒗) =
𝒅
𝒅𝒙(𝒇(𝒙)) ∙ (𝒈((𝒙)))
Put in words:
Find 𝒅𝒚
𝒅𝒙: 𝒚 = (𝒙𝟐 − 𝟕)(𝒙𝟑 + 𝟐)
Check by multiplying first then take the derivative:
The Quotient Rule:
𝒅
𝒅𝒙(
𝒖
𝒗) =
𝒅
𝒅𝒙 (
𝒇(𝒙)
𝒈(𝒙))
Put in words:
Find 𝒇′(𝒙) if 𝒇(𝒙) = 𝒙𝟑
𝒙𝟐− 𝟏
ASSIGNMENT: Page 120 # 1-11 odd, 12-14, 17,18
CALCULUS CHAPTER 3 DERIVATIVES
SECTION 3-3 Product and Quotient Rule (Day 2)
Given the following at x = 2: u(2) = 3, u’(2) = - 4, v(2) = 1, v’(2) = 2.
Find: 𝒅
𝒅𝒙 (𝒖 ∙ 𝒗)
𝒅
𝒅𝒙 (
𝒖
𝒗)
𝒅
𝒅𝒙 (
𝒗
𝒖)
𝒅
𝒅𝒙 (𝟑𝒖 − 𝟐𝒗 + 𝟐𝒖𝒗)
Graph the derivative of the piecewise function below.
ASSIGNMENT: Page 120 #23, 25 - 27, 28(Just the first part), 29 - 32
CALCULUS CHAPTER 3 DERIVATIVES
SECTION 3-4 Velocity
The _____________________ rate of change (slope of a secant) of a function can be found by using the quotients:
m = ________ ____________________ __________________ __________________
A derivative (slope of the tangent) is the _________________________ rate of change.
Definition: f ‘ (x) =
Motion Along a Line (called Linear Motion):
Position function :
Displacement (denoted ) of an object over the time interval [a, b]:
Average Velocity of an object over the time interval [a, b]:
Vave =
Velocity (also called _______________________ ________________) is the derivative of the position function w/r to time.
Ways to denote:
Velocity tells how ___________ and in what ________________ a particle is moving.
An object that is moving forward has a _________________velocity.
An object that is moving backward has a _________________ velocity.
An object that is stopped has a ________________ velocity.
Speed:
Acceleration: Is the _______________________ of the velocity w/r to time. It represents the sudden change in _______________________.
If a particle is speeding up:
If a particle is slowing down:
If a particle is staying the same speed:
Ways to denote:
Jerk: Is the _________________________ of the acceleration. It represents a sudden
change in ______________________________.
Denoted:
A typical linear motion equation: s(t) = - 16t2 + 32t + 120
Ex. A particle moves along a line described by: 𝒔(𝒕) = 𝟏
𝟑𝒕𝟑 − 𝟑𝒕𝟐 + 𝟖𝒕 − 𝟒
a. Find the displacement during the first 6 seconds.
b. Find the vave during the first 6 seconds.
c. Find v(t) and a(t).
d. Find when the particle is stopped.
Marginal Cost: (C’ (x)) The derivative of the cost function. C’(x) represents the cost it takes to make that very part, appliance or piece of equipment.
ASSIGNMENT: Page 129-130 #2 and 3(skip f), 5, 6, 10, 12, 16
CALCULUS CHAPTER 3 DERIVATIVES
SECTION 3-5 Derivatives of Trig Functions
𝒅
𝒅𝒙 sin x =
𝒅
𝒅𝒙 cos x =
𝒅
𝒅𝒙 tan x =
𝒅
𝒅𝒙 csc x =
𝒅
𝒅𝒙 sec x =
𝒅
𝒅𝒙 ctn x =
Proof of derivative of sin x
Examples: Find the derivatives of the following:
𝒚 = 𝟓 𝒄𝒐𝒔 𝒙 𝒇(𝒙) = −𝟐𝝅 𝒄𝒔𝒄 𝒙
𝒚 = 𝒙𝟐 𝒕𝒂𝒏 𝒙 𝒚 = 𝒔𝒊𝒏 𝒙
𝟏+𝒄𝒐𝒔 𝒙
Example: Given the following position function: 𝒔(𝒕) = 𝒔𝒊𝒏 𝒕 + 𝒄𝒐𝒔 𝒕
Find: 𝒗 (𝝅
𝟒) 𝒂 (
𝝅
𝟒)
ASSIGNMENT: Page 140 # 1 – 3, 5, 6, 8, 10, 11, 19, 20, 23 a and b
CALCULUS CHAPTER 3 DERIVATIVES
SECTION 3-6 (Day 1) The Chain Rule
The Chain Rule is for taking derivatives of Compositions of Functions:
𝑰𝒇 𝒚 = (𝒇(𝒈(𝒙))) , 𝒕𝒉𝒆𝒏 𝒚′ = 𝒇′(𝒈(𝒙)) × 𝒈′(𝒙)
It is sometimes called the “Inside-Outside” Rule.
Examples:
𝑭𝒊𝒏𝒅 𝒅𝒚
𝒅𝒙: 𝒚 = (𝟐𝒙𝟑 − 𝟕𝒙 + 𝟑)𝟓
With Trig Functions:
𝑭𝒊𝒏𝒅 𝒅𝒚
𝒅𝒙: 𝒚 = 𝒄𝒐𝒔 (𝟑 𝒙𝟓 − 𝟒𝒙)
A Triple Chain:
𝑭𝒊𝒏𝒅 𝒅𝒚
𝒅𝒙: 𝒚 = 𝒕𝒂𝒏𝟑 𝟔𝒙
In Combos with the Product Rule:
𝑭𝒊𝒏𝒅 𝒅𝒚
𝒅𝒙: 𝒚 = (𝒙𝟐 − 𝟐)𝟑(𝒙𝟑 + 𝟓)−𝟐
ASSIGNMENT: Day 1: Page 146 # 4, 7, 12, 16-18, 20
CALCULUS CHAPTER 3 DERIVATIVES
SECTION 3-6 (Day 2) The Chain Rule
Another way of expressing the Chain Rule:
𝑰𝒇 𝒚 = 𝒇(𝒖) 𝒂𝒏𝒅 𝒖 = 𝒈(𝒙), 𝒕𝒉𝒆𝒏 𝒅𝒚
𝒅𝒙=
𝒅𝒚
𝒅𝒖 ×
𝒅𝒖
𝒅𝒙
Example: Find 𝒅𝒚
𝒅𝒙 if 𝒚 = 𝒄𝒐𝒔 𝒖 𝒂𝒏𝒅 𝒖 = 𝟔𝒙 + 𝟐
1. Composition first, then use the Chain Rule:
2. Use Chain Rule first, then composition.
Derivatives of Parametric Equations:
Recall Parametric Equations: 𝒙 = 𝒔𝒊𝒏 𝟐𝝅 𝒕 and 𝒚 = 𝒄𝒐𝒔 𝟐𝝅 𝒕 𝒇𝒓𝒐𝒎 𝟎 ≤ 𝒕 ≤ 𝟏
The Derivative of Parametric Functions:
𝒅𝒚
𝒅𝒙=
𝒅𝒚𝒅𝒕𝒅𝒙𝒅𝒕
Now, using the parametric equations above, find the equation of the tangent line at 𝒕 =
𝟏
𝟔.
ASSIGNMENT: Page 146 – 147 #22, 29, 40a, 41, 46, 56 a-e
CALCULUS CHAPTER 3 DERIVATIVES
SECTION 3-7 Implicit Differentiation
Explicitly defined functions vs. Implicitly defined functions:
Schema for Implicit Differentiation:
1.
2.
3.
4.
Find 𝒅𝒚
𝒅𝒙: 𝒙𝟐 − 𝟐𝒙𝒚 + 𝒚𝟐 = 𝟕
Now find the equation of the tangent line at (-1, 2)
Find 𝒅𝒚
𝒅𝒙: 𝒄𝒐𝒔 (𝒙 ∙ 𝒚) = 𝟐
Fractional Powers: 𝑭𝒊𝒏𝒅 𝒅𝒚
𝒅𝒙 𝒊𝒇 𝒚 = 𝟑 (𝟐𝒙
−𝟏 𝟐 + 𝟏)
−𝟏 𝟑
ASSIGNMENT: Page 155 5, 7, 9, 19, 21, 23, 27, 30
CALCULUS CHAPTER 3 DERIVATIVES
SECTION 3-8 Derivatives of Inverse Trig Functions
𝒅
𝒅𝒙 (𝒔𝒊𝒏)−𝟏 𝒖 =
𝒅
𝒅𝒙 (𝒄𝒐𝒔)−𝟏 𝒖 =
𝒅
𝒅𝒙 (𝒕𝒂𝒏)−𝟏 𝒖 =
𝒅
𝒅𝒙 (𝒔𝒆𝒄)−𝟏 𝒖 =
𝒅
𝒅𝒙 (𝒄𝒔𝒄)−𝟏 𝒖 =
𝒅
𝒅𝒙 (𝒄𝒕𝒏)−𝟏 𝒖 =
Find 𝒅𝒚
𝒅𝒙: 𝒚 = 𝒔𝒊𝒏−𝟏 (𝟓𝒙𝟐 − 𝟕) =
Find 𝒅𝒚
𝒅𝒙: 𝒚 = 𝒄𝒔𝒄−𝟏 (𝟑𝒙𝟐 + 𝟐𝒙)
Find 𝒅𝒚
𝒅𝒙∶ 𝒚 = 𝒄𝒕𝒏−𝟏 (
𝟏
𝒙) − 𝒕𝒂𝒏−𝟏𝒙
ASSIGNMENT: Page 162 # 7, 9, 12, 17, 19
CALCULUS CHAPTER 3 DERIVATIVES
SECTION 3-9 (Day 1) Derivatives of exponentials and logarithms
The Exponential Function: 𝒅𝒚
𝒅𝒙 𝒆𝒖 =
Find: 𝒅𝒚
𝒅𝒙𝟑𝒙𝟐𝒆𝒙𝟐
=
Any Exponential Function: 𝒅𝒚
𝒅𝒙𝒂𝒖 =
Find: 𝒅𝒚
𝒅𝒙𝟒𝒙𝟐+ 𝟐
The Natural Log Function: 𝒅𝒚
𝒅𝒙𝒍𝒏 𝒖 =
Find: 𝒅𝒚
𝒅𝒙 𝒍𝒏 (𝟑𝒙𝟐)
𝒅𝒚
𝒅𝒙 ((𝒙)(𝒍𝒏)𝒙) =
ASSIGNMENT: Page 170 #4 – 10, 15 – 17, 21, 23, 28 – 30
CALCULUS CHAPTER 3 DERIVATIVES
SECTION 3-9 (Day 2) Derivatives of Logarithms and Log Differentiation
Derivatives of regular logarithms: 𝒅𝒚
𝒅𝒙 𝐥𝐨𝐠𝒂 𝒖 =
Find 𝒅𝒚
𝒅𝒙: 𝒚 = 𝐥𝐨𝐠𝟏𝟎 √𝒙 + 𝟏
Logarithmic Differentiation:
Basic Steps: 1. Take ln of both sides of the equation.
2. Break down using log rules.
3. Take the derivative of both sides.
4. Solve equation for 𝒅𝒚
𝒅𝒙.
Find 𝒅𝒚
𝒅𝒙 𝒐𝒇: 𝒚 = 𝒙𝒔𝒊𝒏 𝒙 𝒚 = (𝒔𝒊𝒏 𝒙)𝒙
ASSIGNMENT: Page 170 #12, 19, 31, 35, 38, 39, 46
DERIVATIVE FORMULAS FOR CHAPTER THREE
𝒅
𝒅𝒙 𝒖𝒏 =
𝒅
𝒅𝒙 𝒖 ∙ 𝒗 =
𝒅
𝒅𝒙 𝒆𝒖 =
𝒅
𝒅𝒙 𝒍𝒏 𝒖 =
𝒅
𝒅𝒙 𝒂𝒖 =
𝒅
𝒅𝒙 𝐥𝐨𝐠𝒂 𝒖 =
𝒅
𝒅𝒙 𝒔𝒊𝒏 𝒖 =
𝒅
𝒅𝒙 𝒄𝒐𝒔 𝒖 =
𝒅
𝒅𝒙 𝒕𝒂𝒏 𝒖 =
𝒅
𝒅𝒙 𝒄𝒕𝒏 𝒖 =
𝒅
𝒅𝒙 𝒔𝒆𝒄 𝒖 =
𝒅
𝒅𝒙 𝒄𝒔𝒄 𝒖 =
𝒅
𝒅𝒙 𝒔𝒊𝒏−𝟏 𝒖 =
𝒅
𝒅𝒙 𝒄𝒐𝒔−𝟏 𝒖 =
𝒅
𝒅𝒙 𝒕𝒂𝒏−𝟏 𝒖 =
𝒅
𝒅𝒙 𝒄𝒕𝒏−𝟏 𝒖 =
𝒅
𝒅𝒙 𝒔𝒆𝒄−𝟏 𝒖 =
𝒅
𝒅𝒙 𝒄𝒔𝒄−𝟏 𝒖 =
Parametric equations: 𝒅𝒚
𝒅𝒙=
CALCULUS CHAPTER THREE ASSIGNMENTS
SECTION 3-1 DERIVATIVES
ASSIGNMENT: Page 101 – 102 #1, 4-12, 15-17
SECTION 3-2 Differentiability
ASSIGNMENT: Page 111 #1-10
SECTION 3-3 Product and Quotient Rule (Day 1)
ASSIGNMENT: Page 120 # 1-11 odd, 12-14, 17,18
SECTION 3-3 Product and Quotient Rule (Day 2)
ASSIGNMENT: Page 120 #23, 25 - 27, 28(Just the first part), 29 - 32
SECTION 3-4 Velocity
ASSIGNMENT: Page 129-130 #2 and 3(skip f), 5, 6, 10, 12, 16
REVIEW WORKSHEET
CHAPTER THREE QUIZ #1
SECTION 3-5 Derivatives of Trig Functions
ASSIGNMENT: Page 140 # 1 – 3, 5, 6, 8, 10, 11, 19, 20, 23 a and b
SECTION 3-6 (Day 1) The Chain Rule
ASSIGNMENT: Page 146 # 4, 7, 12, 16-18, 20
SECTION 3-6 (Day 2) The Chain Rule
ASSIGNMENT: Page 146 – 147 #22, 29, 40a, 41, 46, 56 a-e
SECTION 3-7 Implicit Differentiation
ASSIGNMENT: Page 155 5, 7, 9, 19, 21, 23, 27, 30
REVIEW WORKSHEET
CHAPTER THREE QUIZ #2
SECTION 3-8 Derivatives of Inverse Trig Functions
ASSIGNMENT: Page 162 # 7, 9, 12, 17, 19
SECTION 3-9 (Day 1) Derivatives of Exponentials and Logarithms
ASSIGNMENT: Page 170 #4 – 10, 15 – 17, 21, 23, 28 – 30
SECTION 3-9 (Day 2) Derivatives of Logarithms and Log Differentiation
ASSIGNMENT: Page 170 #12, 19, 31, 35, 38, 39, 46
REVIEW WORKSHEET
CHAPTER THREE QUIZ #3