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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

Implicit Differentiation

Slide Duration:

Table of Contents

I. Limits and Derivatives
Overview & Slopes of Curves

42m 8s

Intro
0:00
Overview & Slopes of Curves
0:21
Differential and Integral
0:22
Fundamental Theorem of Calculus
6:36
Differentiation or Taking the Derivative
14:24
What Does the Derivative Mean and How do We Find it?
15:18
Example: f'(x)
19:24
Example: f(x) = sin (x)
29:16
General Procedure for Finding the Derivative of f(x)
37:33
More on Slopes of Curves

50m 53s

Intro
0:00
Slope of the Secant Line along a Curve
0:12
Slope of the Tangent Line to f(x) at a Particlar Point
0:13
Slope of the Secant Line along a Curve
2:59
Instantaneous Slope
6:51
Instantaneous Slope
6:52
Example: Distance, Time, Velocity
13:32
Instantaneous Slope and Average Slope
25:42
Slope & Rate of Change
29:55
Slope & Rate of Change
29:56
Example: Slope = 2
33:16
Example: Slope = 4/3
34:32
Example: Slope = 4 (m/s)
39:12
Example: Density = Mass / Volume
40:33
Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change
47:46
Example Problems for Slopes of Curves

59m 12s

Intro
0:00
Example I: Water Tank
0:13
Part A: Which is the Independent Variable and Which is the Dependent?
2:00
Part B: Average Slope
3:18
Part C: Express These Slopes as Rates-of-Change
9:28
Part D: Instantaneous Slope
14:54
Example II: y = √(x-3)
28:26
Part A: Calculate the Slope of the Secant Line
30:39
Part B: Instantaneous Slope
41:26
Part C: Equation for the Tangent Line
43:59
Example III: Object in the Air
49:37
Part A: Average Velocity
50:37
Part B: Instantaneous Velocity
55:30
Desmos Tutorial

18m 43s

Intro
0:00
Desmos Tutorial
1:42
Desmos Tutorial
1:43
Things You Must Learn To Do on Your Particular Calculator
2:39
Things You Must Learn To Do on Your Particular Calculator
2:40
Example I: y=sin x
4:54
Example II: y=x³ and y = d/(dx) (x³)
9:22
Example III: y = x² {-5 <= x <= 0} and y = cos x {0 < x < 6}
13:15
The Limit of a Function

51m 53s

Intro
0:00
The Limit of a Function
0:14
The Limit of a Function
0:15
Graph: Limit of a Function
12:24
Table of Values
16:02
lim x→a f(x) Does not Say What Happens When x = a
20:05
Example I: f(x) = x²
24:34
Example II: f(x) = 7
27:05
Example III: f(x) = 4.5
30:33
Example IV: f(x) = 1/x
34:03
Example V: f(x) = 1/x²
36:43
The Limit of a Function, Cont.
38:16
Infinity and Negative Infinity
38:17
Does Not Exist
42:45
Summary
46:48
Example Problems for the Limit of a Function

24m 43s

Intro
0:00
Example I: Explain in Words What the Following Symbols Mean
0:10
Example II: Find the Following Limit
5:21
Example III: Use the Graph to Find the Following Limits
7:35
Example IV: Use the Graph to Find the Following Limits
11:48
Example V: Sketch the Graph of a Function that Satisfies the Following Properties
15:25
Example VI: Find the Following Limit
18:44
Example VII: Find the Following Limit
20:06
Calculating Limits Mathematically

53m 48s

Intro
0:00
Plug-in Procedure
0:09
Plug-in Procedure
0:10
Limit Laws
9:14
Limit Law 1
10:05
Limit Law 2
10:54
Limit Law 3
11:28
Limit Law 4
11:54
Limit Law 5
12:24
Limit Law 6
13:14
Limit Law 7
14:38
Plug-in Procedure, Cont.
16:35
Plug-in Procedure, Cont.
16:36
Example I: Calculating Limits Mathematically
20:50
Example II: Calculating Limits Mathematically
27:37
Example III: Calculating Limits Mathematically
31:42
Example IV: Calculating Limits Mathematically
35:36
Example V: Calculating Limits Mathematically
40:58
Limits Theorem
44:45
Limits Theorem 1
44:46
Limits Theorem 2: Squeeze Theorem
46:34
Example VI: Calculating Limits Mathematically
49:26
Example Problems for Calculating Limits Mathematically

21m 22s

Intro
0:00
Example I: Evaluate the Following Limit by Showing Each Application of a Limit Law
0:16
Example II: Evaluate the Following Limit
1:51
Example III: Evaluate the Following Limit
3:36
Example IV: Evaluate the Following Limit
8:56
Example V: Evaluate the Following Limit
11:19
Example VI: Calculating Limits Mathematically
13:19
Example VII: Calculating Limits Mathematically
14:59
Calculating Limits as x Goes to Infinity

50m 1s

Intro
0:00
Limit as x Goes to Infinity
0:14
Limit as x Goes to Infinity
0:15
Let's Look at f(x) = 1 / (x-3)
1:04
Summary
9:34
Example I: Calculating Limits as x Goes to Infinity
12:16
Example II: Calculating Limits as x Goes to Infinity
21:22
Example III: Calculating Limits as x Goes to Infinity
24:10
Example IV: Calculating Limits as x Goes to Infinity
36:00
Example Problems for Limits at Infinity

36m 31s

Intro
0:00
Example I: Calculating Limits as x Goes to Infinity
0:14
Example II: Calculating Limits as x Goes to Infinity
3:27
Example III: Calculating Limits as x Goes to Infinity
8:11
Example IV: Calculating Limits as x Goes to Infinity
14:20
Example V: Calculating Limits as x Goes to Infinity
20:07
Example VI: Calculating Limits as x Goes to Infinity
23:36
Continuity

53m

Intro
0:00
Definition of Continuity
0:08
Definition of Continuity
0:09
Example: Not Continuous
3:52
Example: Continuous
4:58
Example: Not Continuous
5:52
Procedure for Finding Continuity
9:45
Law of Continuity
13:44
Law of Continuity
13:45
Example I: Determining Continuity on a Graph
15:55
Example II: Show Continuity & Determine the Interval Over Which the Function is Continuous
17:57
Example III: Is the Following Function Continuous at the Given Point?
22:42
Theorem for Composite Functions
25:28
Theorem for Composite Functions
25:29
Example IV: Is cos(x³ + ln x) Continuous at x=π/2?
27:00
Example V: What Value of A Will make the Following Function Continuous at Every Point of Its Domain?
34:04
Types of Discontinuity
39:18
Removable Discontinuity
39:33
Jump Discontinuity
40:06
Infinite Discontinuity
40:32
Intermediate Value Theorem
40:58
Intermediate Value Theorem: Hypothesis & Conclusion
40:59
Intermediate Value Theorem: Graphically
43:40
Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6]
47:46
Derivative I

40m 2s

Intro
0:00
Derivative
0:09
Derivative
0:10
Example I: Find the Derivative of f(x)=x³
2:20
Notations for the Derivative
7:32
Notations for the Derivative
7:33
Derivative & Rate of Change
11:14
Recall the Rate of Change
11:15
Instantaneous Rate of Change
17:04
Graphing f(x) and f'(x)
19:10
Example II: Find the Derivative of x⁴ - x²
24:00
Example III: Find the Derivative of f(x)=√x
30:51
Derivatives II

53m 45s

Intro
0:00
Example I: Find the Derivative of (2+x)/(3-x)
0:18
Derivatives II
9:02
f(x) is Differentiable if f'(x) Exists
9:03
Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other
17:19
Geometrically: Differentiability Means the Graph is Smooth
18:44
Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=0
20:53
Example II: For x > 0
23:53
Example II: For x < 0
25:36
Example II: What is f(0) and What is the lim |x| as x→0?
30:46
Differentiability & Continuity
34:22
Differentiability & Continuity
34:23
How Can a Function Not be Differentiable at a Point?
39:38
How Can a Function Not be Differentiable at a Point?
39:39
Higher Derivatives
41:58
Higher Derivatives
41:59
Derivative Operator
45:12
Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³
49:29
More Example Problems for The Derivative

31m 38s

Intro
0:00
Example I: Sketch f'(x)
0:10
Example II: Sketch f'(x)
2:14
Example III: Find the Derivative of the Following Function sing the Definition
3:49
Example IV: Determine f, f', and f'' on a Graph
12:43
Example V: Find an Equation for the Tangent Line to the Graph of the Following Function at the Given x-value
13:40
Example VI: Distance vs. Time
20:15
Example VII: Displacement, Velocity, and Acceleration
23:56
Example VIII: Graph the Displacement Function
28:20
II. Differentiation
Differentiation of Polynomials & Exponential Functions

47m 35s

Intro
0:00
Differentiation of Polynomials & Exponential Functions
0:15
Derivative of a Function
0:16
Derivative of a Constant
2:35
Power Rule
3:08
If C is a Constant
4:19
Sum Rule
5:22
Exponential Functions
6:26
Example I: Differentiate
7:45
Example II: Differentiate
12:38
Example III: Differentiate
15:13
Example IV: Differentiate
16:20
Example V: Differentiate
19:19
Example VI: Find the Equation of the Tangent Line to a Function at a Given Point
12:18
Example VII: Find the First & Second Derivatives
25:59
Example VIII
27:47
Part A: Find the Velocity & Acceleration Functions as Functions of t
27:48
Part B: Find the Acceleration after 3 Seconds
30:12
Part C: Find the Acceleration when the Velocity is 0
30:53
Part D: Graph the Position, Velocity, & Acceleration Graphs
32:50
Example IX: Find a Cubic Function Whose Graph has Horizontal Tangents
34:53
Example X: Find a Point on a Graph
42:31
The Product, Power & Quotient Rules

47m 25s

Intro
0:00
The Product, Power and Quotient Rules
0:19
Differentiate Functions
0:20
Product Rule
5:30
Quotient Rule
9:15
Power Rule
10:00
Example I: Product Rule
13:48
Example II: Quotient Rule
16:13
Example III: Power Rule
18:28
Example IV: Find dy/dx
19:57
Example V: Find dy/dx
24:53
Example VI: Find dy/dx
28:38
Example VII: Find an Equation for the Tangent to the Curve
34:54
Example VIII: Find d²y/dx²
38:08
Derivatives of the Trigonometric Functions

41m 8s

Intro
0:00
Derivatives of the Trigonometric Functions
0:09
Let's Find the Derivative of f(x) = sin x
0:10
Important Limits to Know
4:59
d/dx (sin x)
6:06
d/dx (cos x)
6:38
d/dx (tan x)
6:50
d/dx (csc x)
7:02
d/dx (sec x)
7:15
d/dx (cot x)
7:27
Example I: Differentiate f(x) = x² - 4 cos x
7:56
Example II: Differentiate f(x) = x⁵ tan x
9:04
Example III: Differentiate f(x) = (cos x) / (3 + sin x)
10:56
Example IV: Differentiate f(x) = e^x / (tan x - sec x)
14:06
Example V: Differentiate f(x) = (csc x - 4) / (cot x)
15:37
Example VI: Find an Equation of the Tangent Line
21:48
Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent?
25:17
Example VIII: Ladder Problem
28:23
Example IX: Evaluate
33:22
Example X: Evaluate
36:38
The Chain Rule

24m 56s

Intro
0:00
The Chain Rule
0:13
Recall the Composite Functions
0:14
Derivatives of Composite Functions
1:34
Example I: Identify f(x) and g(x) and Differentiate
6:41
Example II: Identify f(x) and g(x) and Differentiate
9:47
Example III: Differentiate
11:03
Example IV: Differentiate f(x) = -5 / (x² + 3)³
12:15
Example V: Differentiate f(x) = cos(x² + c²)
14:35
Example VI: Differentiate f(x) = cos⁴x +c²
15:41
Example VII: Differentiate
17:03
Example VIII: Differentiate f(x) = sin(tan x²)
19:01
Example IX: Differentiate f(x) = sin(tan² x)
21:02
More Chain Rule Example Problems

25m 32s

Intro
0:00
Example I: Differentiate f(x) = sin(cos(tanx))
0:38
Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point
2:25
Example III: F(x) = f(g(x)), Find F' (6)
4:22
Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window
5:35
Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴
10:18
Example VI: Differentiate f(x) = sec²(12x)
12:28
Example VII: Differentiate
14:41
Example VIII: Differentiate
19:25
Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time
21:13
Implicit Differentiation

52m 31s

Intro
0:00
Implicit Differentiation
0:09
Implicit Differentiation
0:10
Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y
12:15
Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 14
19:18
Example III: Find (dy)/(dx) of x³y² + y³x² = 4x
21:43
Example IV: Find (dy)/(dx) of the Following Equation
24:13
Example V: Find (dy)/(dx) of 6sin x cos y = 1
29:00
Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y
31:02
Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x
37:36
Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²-y²)
41:03
Example IX: Find (d²y)/(dx²) of x² + y² = 25
44:05
Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x)
47:48
III. Applications of the Derivative
Linear Approximations & Differentials

47m 34s

Intro
0:00
Linear Approximations & Differentials
0:09
Linear Approximations & Differentials
0:10
Example I: Linear Approximations & Differentials
11:27
Example II: Linear Approximations & Differentials
20:19
Differentials
30:32
Differentials
30:33
Example III: Linear Approximations & Differentials
34:09
Example IV: Linear Approximations & Differentials
35:57
Example V: Relative Error
38:46
Related Rates

45m 33s

Intro
0:00
Related Rates
0:08
Strategy for Solving Related Rates Problems #1
0:09
Strategy for Solving Related Rates Problems #2
1:46
Strategy for Solving Related Rates Problems #3
2:06
Strategy for Solving Related Rates Problems #4
2:50
Strategy for Solving Related Rates Problems #5
3:38
Example I: Radius of a Balloon
5:15
Example II: Ladder
12:52
Example III: Water Tank
19:08
Example IV: Distance between Two Cars
29:27
Example V: Line-of-Sight
36:20
More Related Rates Examples

37m 17s

Intro
0:00
Example I: Shadow
0:14
Example II: Particle
4:45
Example III: Water Level
10:28
Example IV: Clock
20:47
Example V: Distance between a House and a Plane
29:11
Maximum & Minimum Values of a Function

40m 44s

Intro
0:00
Maximum & Minimum Values of a Function, Part 1
0:23
Absolute Maximum
2:20
Absolute Minimum
2:52
Local Maximum
3:38
Local Minimum
4:26
Maximum & Minimum Values of a Function, Part 2
6:11
Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min
7:18
Function with Local Max & Min but No Absolute Max & Min
8:48
Formal Definitions
10:43
Absolute Maximum
11:18
Absolute Minimum
12:57
Local Maximum
14:37
Local Minimum
16:25
Extreme Value Theorem
18:08
Theorem: f'(c) = 0
24:40
Critical Number (Critical Value)
26:14
Procedure for Finding the Critical Values of f(x)
28:32
Example I: Find the Critical Values of f(x) x + sinx
29:51
Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π]
35:31
Example Problems for Max & Min

40m 44s

Intro
0:00
Example I: Identify Absolute and Local Max & Min on the Following Graph
0:11
Example II: Sketch the Graph of a Continuous Function
3:11
Example III: Sketch the Following Graphs
4:40
Example IV: Find the Critical Values of f (x) = 3x⁴ - 7x³ + 4x²
6:13
Example V: Find the Critical Values of f(x) = |2x - 5|
8:42
Example VI: Find the Critical Values
11:42
Example VII: Find the Critical Values f(x) = cos²(2x) on [0,2π]
16:57
Example VIII: Find the Absolute Max & Min f(x) = 2sinx + 2cos x on [0,(π/3)]
20:08
Example IX: Find the Absolute Max & Min f(x) = (ln(2x)) / x on [1,3]
24:39
The Mean Value Theorem

25m 54s

Intro
0:00
Rolle's Theorem
0:08
Rolle's Theorem: If & Then
0:09
Rolle's Theorem: Geometrically
2:06
There May Be More than 1 c Such That f'( c ) = 0
3:30
Example I: Rolle's Theorem
4:58
The Mean Value Theorem
9:12
The Mean Value Theorem: If & Then
9:13
The Mean Value Theorem: Geometrically
11:07
Example II: Mean Value Theorem
13:43
Example III: Mean Value Theorem
21:19
Using Derivatives to Graph Functions, Part I

25m 54s

Intro
0:00
Using Derivatives to Graph Functions, Part I
0:12
Increasing/ Decreasing Test
0:13
Example I: Find the Intervals Over Which the Function is Increasing & Decreasing
3:26
Example II: Find the Local Maxima & Minima of the Function
19:18
Example III: Find the Local Maxima & Minima of the Function
31:39
Using Derivatives to Graph Functions, Part II

44m 58s

Intro
0:00
Using Derivatives to Graph Functions, Part II
0:13
Concave Up & Concave Down
0:14
What Does This Mean in Terms of the Derivative?
6:14
Point of Inflection
8:52
Example I: Graph the Function
13:18
Example II: Function x⁴ - 5x²
19:03
Intervals of Increase & Decrease
19:04
Local Maxes and Mins
25:01
Intervals of Concavity & X-Values for the Points of Inflection
29:18
Intervals of Concavity & Y-Values for the Points of Inflection
34:18
Graphing the Function
40:52
Example Problems I

49m 19s

Intro
0:00
Example I: Intervals, Local Maxes & Mins
0:26
Example II: Intervals, Local Maxes & Mins
5:05
Example III: Intervals, Local Maxes & Mins, and Inflection Points
13:40
Example IV: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
23:02
Example V: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
34:36
Example Problems III

59m 1s

Intro
0:00
Example I: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
0:11
Example II: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
21:24
Example III: Cubic Equation f(x) = Ax³ + Bx² + Cx + D
37:56
Example IV: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
46:19
L'Hospital's Rule

30m 9s

Intro
0:00
L'Hospital's Rule
0:19
Indeterminate Forms
0:20
L'Hospital's Rule
3:38
Example I: Evaluate the Following Limit Using L'Hospital's Rule
8:50
Example II: Evaluate the Following Limit Using L'Hospital's Rule
10:30
Indeterminate Products
11:54
Indeterminate Products
11:55
Example III: L'Hospital's Rule & Indeterminate Products
13:57
Indeterminate Differences
17:00
Indeterminate Differences
17:01
Example IV: L'Hospital's Rule & Indeterminate Differences
18:57
Indeterminate Powers
22:20
Indeterminate Powers
22:21
Example V: L'Hospital's Rule & Indeterminate Powers
25:13
Example Problems for L'Hospital's Rule

38m 14s

Intro
0:00
Example I: Evaluate the Following Limit
0:17
Example II: Evaluate the Following Limit
2:45
Example III: Evaluate the Following Limit
6:54
Example IV: Evaluate the Following Limit
8:43
Example V: Evaluate the Following Limit
11:01
Example VI: Evaluate the Following Limit
14:48
Example VII: Evaluate the Following Limit
17:49
Example VIII: Evaluate the Following Limit
20:37
Example IX: Evaluate the Following Limit
25:16
Example X: Evaluate the Following Limit
32:44
Optimization Problems I

49m 59s

Intro
0:00
Example I: Find the Dimensions of the Box that Gives the Greatest Volume
1:23
Fundamentals of Optimization Problems
18:08
Fundamental #1
18:33
Fundamental #2
19:09
Fundamental #3
19:19
Fundamental #4
20:59
Fundamental #5
21:55
Fundamental #6
23:44
Example II: Demonstrate that of All Rectangles with a Given Perimeter, the One with the Largest Area is a Square
24:36
Example III: Find the Points on the Ellipse 9x² + y² = 9 Farthest Away from the Point (1,0)
35:13
Example IV: Find the Dimensions of the Rectangle of Largest Area that can be Inscribed in a Circle of Given Radius R
43:10
Optimization Problems II

55m 10s

Intro
0:00
Example I: Optimization Problem
0:13
Example II: Optimization Problem
17:34
Example III: Optimization Problem
35:06
Example IV: Revenue, Cost, and Profit
43:22
Newton's Method

30m 22s

Intro
0:00
Newton's Method
0:45
Newton's Method
0:46
Example I: Find x2 and x3
13:18
Example II: Use Newton's Method to Approximate
15:48
Example III: Find the Root of the Following Equation to 6 Decimal Places
19:57
Example IV: Use Newton's Method to Find the Coordinates of the Inflection Point
23:11
IV. Integrals
Antiderivatives

55m 26s

Intro
0:00
Antiderivatives
0:23
Definition of an Antiderivative
0:24
Antiderivative Theorem
7:58
Function & Antiderivative
12:10
x^n
12:30
1/x
13:00
e^x
13:08
cos x
13:18
sin x
14:01
sec² x
14:11
secxtanx
14:18
1/√(1-x²)
14:26
1/(1+x²)
14:36
-1/√(1-x²)
14:45
Example I: Find the Most General Antiderivative for the Following Functions
15:07
Function 1: f(x) = x³ -6x² + 11x - 9
15:42
Function 2: f(x) = 14√(x) - 27 4√x
19:12
Function 3: (fx) = cos x - 14 sinx
20:53
Function 4: f(x) = (x⁵+2√x )/( x^(4/3) )
22:10
Function 5: f(x) = (3e^x) - 2/(1+x²)
25:42
Example II: Given the Following, Find the Original Function f(x)
26:37
Function 1: f'(x) = 5x³ - 14x + 24, f(2) = 40
27:55
Function 2: f'(x) 3 sinx + sec²x, f(π/6) = 5
30:34
Function 3: f''(x) = 8x - cos x, f(1.5) = 12.7, f'(1.5) = 4.2
32:54
Function 4: f''(x) = 5/(√x), f(2) 15, f'(2) = 7
37:54
Example III: Falling Object
41:58
Problem 1: Find an Equation for the Height of the Ball after t Seconds
42:48
Problem 2: How Long Will It Take for the Ball to Strike the Ground?
48:30
Problem 3: What is the Velocity of the Ball as it Hits the Ground?
49:52
Problem 4: Initial Velocity of 6 m/s, How Long Does It Take to Reach the Ground?
50:46
The Area Under a Curve

51m 3s

Intro
0:00
The Area Under a Curve
0:13
Approximate Using Rectangles
0:14
Let's Do This Again, Using 4 Different Rectangles
9:40
Approximate with Rectangles
16:10
Left Endpoint
18:08
Right Endpoint
25:34
Left Endpoint vs. Right Endpoint
30:58
Number of Rectangles
34:08
True Area
37:36
True Area
37:37
Sigma Notation & Limits
43:32
When You Have to Explicitly Solve Something
47:56
Example Problems for Area Under a Curve

33m 7s

Intro
0:00
Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve
0:10
Example II: Using 5 Rectangles, Approximate the Area Under the Curve
11:32
Example III: Find the True Area by Evaluating the Limit Expression
16:07
Example IV: Find the True Area by Evaluating the Limit Expression
24:52
The Definite Integral

43m 19s

Intro
0:00
The Definite Integral
0:08
Definition to Find the Area of a Curve
0:09
Definition of the Definite Integral
4:08
Symbol for Definite Integral
8:45
Regions Below the x-axis
15:18
Associating Definite Integral to a Function
19:38
Integrable Function
27:20
Evaluating the Definite Integral
29:26
Evaluating the Definite Integral
29:27
Properties of the Definite Integral
35:24
Properties of the Definite Integral
35:25
Example Problems for The Definite Integral

32m 14s

Intro
0:00
Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals
0:11
Example II: Express the Following Limit as a Definite Integral
5:28
Example III: Evaluate the Following Definite Integral Using the Definition
6:28
Example IV: Evaluate the Following Integral Using the Definition
17:06
Example V: Evaluate the Following Definite Integral by Using Areas
25:41
Example VI: Definite Integral
30:36
The Fundamental Theorem of Calculus

24m 17s

Intro
0:00
The Fundamental Theorem of Calculus
0:17
Evaluating an Integral
0:18
Lim as x → ∞
12:19
Taking the Derivative
14:06
Differentiation & Integration are Inverse Processes
15:04
1st Fundamental Theorem of Calculus
20:08
1st Fundamental Theorem of Calculus
20:09
2nd Fundamental Theorem of Calculus
22:30
2nd Fundamental Theorem of Calculus
22:31
Example Problems for the Fundamental Theorem

25m 21s

Intro
0:00
Example I: Find the Derivative of the Following Function
0:17
Example II: Find the Derivative of the Following Function
1:40
Example III: Find the Derivative of the Following Function
2:32
Example IV: Find the Derivative of the Following Function
5:55
Example V: Evaluate the Following Integral
7:13
Example VI: Evaluate the Following Integral
9:46
Example VII: Evaluate the Following Integral
12:49
Example VIII: Evaluate the Following Integral
13:53
Example IX: Evaluate the Following Graph
15:24
Local Maxs and Mins for g(x)
15:25
Where Does g(x) Achieve Its Absolute Max on [0,8]
20:54
On What Intervals is g(x) Concave Up/Down?
22:20
Sketch a Graph of g(x)
24:34
More Example Problems, Including Net Change Applications

34m 22s

Intro
0:00
Example I: Evaluate the Following Indefinite Integral
0:10
Example II: Evaluate the Following Definite Integral
0:59
Example III: Evaluate the Following Integral
2:59
Example IV: Velocity Function
7:46
Part A: Net Displacement
7:47
Part B: Total Distance Travelled
13:15
Example V: Linear Density Function
20:56
Example VI: Acceleration Function
25:10
Part A: Velocity Function at Time t
25:11
Part B: Total Distance Travelled During the Time Interval
28:38
Solving Integrals by Substitution

27m 20s

Intro
0:00
Table of Integrals
0:35
Example I: Evaluate the Following Indefinite Integral
2:02
Example II: Evaluate the Following Indefinite Integral
7:27
Example IIII: Evaluate the Following Indefinite Integral
10:57
Example IV: Evaluate the Following Indefinite Integral
12:33
Example V: Evaluate the Following
14:28
Example VI: Evaluate the Following
16:00
Example VII: Evaluate the Following
19:01
Example VIII: Evaluate the Following
21:49
Example IX: Evaluate the Following
24:34
V. Applications of Integration
Areas Between Curves

34m 56s

Intro
0:00
Areas Between Two Curves: Function of x
0:08
Graph 1: Area Between f(x) & g(x)
0:09
Graph 2: Area Between f(x) & g(x)
4:07
Is It Possible to Write as a Single Integral?
8:20
Area Between the Curves on [a,b]
9:24
Absolute Value
10:32
Formula for Areas Between Two Curves: Top Function - Bottom Function
17:03
Areas Between Curves: Function of y
17:49
What if We are Given Functions of y?
17:50
Formula for Areas Between Two Curves: Right Function - Left Function
21:48
Finding a & b
22:32
Example Problems for Areas Between Curves

42m 55s

Intro
0:00
Instructions for the Example Problems
0:10
Example I: y = 7x - x² and y=x
0:37
Example II: x=y²-3, x=e^((1/2)y), y=-1, and y=2
6:25
Example III: y=(1/x), y=(1/x³), and x=4
12:25
Example IV: 15-2x² and y=x²-5
15:52
Example V: x=(1/8)y³ and x=6-y²
20:20
Example VI: y=cos x, y=sin(2x), [0,π/2]
24:34
Example VII: y=2x², y=10x², 7x+2y=10
29:51
Example VIII: Velocity vs. Time
33:23
Part A: At 2.187 Minutes, Which care is Further Ahead?
33:24
Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent?
36:32
Part C: At 4 Minutes Which Car is Ahead?
37:11
Part D: At What Time Will the Cars be Side by Side?
37:50
Volumes I: Slices

34m 15s

Intro
0:00
Volumes I: Slices
0:18
Rotate the Graph of y=√x about the x-axis
0:19
How can I use Integration to Find the Volume?
3:16
Slice the Solid Like a Loaf of Bread
5:06
Volumes Definition
8:56
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
12:18
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
19:05
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
25:28
Volumes II: Volumes by Washers

51m 43s

Intro
0:00
Volumes II: Volumes by Washers
0:11
Rotating Region Bounded by y=x³ & y=x around the x-axis
0:12
Equation for Volumes by Washer
11:14
Process for Solving Volumes by Washer
13:40
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
15:58
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
25:07
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
34:20
Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
44:05
Volumes III: Solids That Are Not Solids-of-Revolution

49m 36s

Intro
0:00
Solids That Are Not Solids-of-Revolution
0:11
Cross-Section Area Review
0:12
Cross-Sections That Are Not Solids-of-Revolution
7:36
Example I: Find the Volume of a Pyramid Whose Base is a Square of Side-length S, and Whose Height is H
10:54
Example II: Find the Volume of a Solid Whose Cross-sectional Areas Perpendicular to the Base are Equilateral Triangles
20:39
Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of Side-Length A, and Whose Height is H
29:27
Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 64
36:47
Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3-x² and the x-axis
46:13
Volumes IV: Volumes By Cylindrical Shells

50m 2s

Intro
0:00
Volumes by Cylindrical Shells
0:11
Find the Volume of the Following Region
0:12
Volumes by Cylindrical Shells: Integrating Along x
14:12
Volumes by Cylindrical Shells: Integrating Along y
14:40
Volumes by Cylindrical Shells Formulas
16:22
Example I: Using the Method of Cylindrical Shells, Find the Volume of the Solid
18:33
Example II: Using the Method of Cylindrical Shells, Find the Volume of the Solid
25:57
Example III: Using the Method of Cylindrical Shells, Find the Volume of the Solid
31:38
Example IV: Using the Method of Cylindrical Shells, Find the Volume of the Solid
38:44
Example V: Using the Method of Cylindrical Shells, Find the Volume of the Solid
44:03
The Average Value of a Function

32m 13s

Intro
0:00
The Average Value of a Function
0:07
Average Value of f(x)
0:08
What if The Domain of f(x) is Not Finite?
2:23
Let's Calculate Average Value for f(x) = x² [2,5]
4:46
Mean Value Theorem for Integrate
9:25
Example I: Find the Average Value of the Given Function Over the Given Interval
14:06
Example II: Find the Average Value of the Given Function Over the Given Interval
18:25
Example III: Find the Number A Such that the Average Value of the Function f(x) = -4x² + 8x + 4 Equals 2 Over the Interval [-1,A]
24:04
Example IV: Find the Average Density of a Rod
27:47
VI. Techniques of Integration
Integration by Parts

50m 32s

Intro
0:00
Integration by Parts
0:08
The Product Rule for Differentiation
0:09
Integrating Both Sides Retains the Equality
0:52
Differential Notation
2:24
Example I: ∫ x cos x dx
5:41
Example II: ∫ x² sin(2x)dx
12:01
Example III: ∫ (e^x) cos x dx
18:19
Example IV: ∫ (sin^-1) (x) dx
23:42
Example V: ∫₁⁵ (lnx)² dx
28:25
Summary
32:31
Tabular Integration
35:08
Case 1
35:52
Example: ∫x³sinx dx
36:39
Case 2
40:28
Example: ∫e^(2x) sin 3x
41:14
Trigonometric Integrals I

24m 50s

Intro
0:00
Example I: ∫ sin³ (x) dx
1:36
Example II: ∫ cos⁵(x)sin²(x)dx
4:36
Example III: ∫ sin⁴(x)dx
9:23
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sin^m) (x) (cos^p) (x) dx
15:59
#1: Power of sin is Odd
16:00
#2: Power of cos is Odd
16:41
#3: Powers of Both sin and cos are Odd
16:55
#4: Powers of Both sin and cos are Even
17:10
Example IV: ∫ tan⁴ (x) sec⁴ (x) dx
17:34
Example V: ∫ sec⁹(x) tan³(x) dx
20:55
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sec^m) (x) (tan^p) (x) dx
23:31
#1: Power of sec is Odd
23:32
#2: Power of tan is Odd
24:04
#3: Powers of sec is Odd and/or Power of tan is Even
24:18
Trigonometric Integrals II

22m 12s

Intro
0:00
Trigonometric Integrals II
0:09
Recall: ∫tanx dx
0:10
Let's Find ∫secx dx
3:23
Example I: ∫ tan⁵ (x) dx
6:23
Example II: ∫ sec⁵ (x) dx
11:41
Summary: How to Deal with Integrals of Different Types
19:04
Identities to Deal with Integrals of Different Types
19:05
Example III: ∫cos(5x)sin(9x)dx
19:57
More Example Problems for Trigonometric Integrals

17m 22s

Intro
0:00
Example I: ∫sin²(x)cos⁷(x)dx
0:14
Example II: ∫x sin²(x) dx
3:56
Example III: ∫csc⁴ (x/5)dx
8:39
Example IV: ∫( (1-tan²x)/(sec²x) ) dx
11:17
Example V: ∫ 1 / (sinx-1) dx
13:19
Integration by Partial Fractions I

55m 12s

Intro
0:00
Integration by Partial Fractions I
0:11
Recall the Idea of Finding a Common Denominator
0:12
Decomposing a Rational Function to Its Partial Fractions
4:10
2 Types of Rational Function: Improper & Proper
5:16
Improper Rational Function
7:26
Improper Rational Function
7:27
Proper Rational Function
11:16
Proper Rational Function & Partial Fractions
11:17
Linear Factors
14:04
Irreducible Quadratic Factors
15:02
Case 1: G(x) is a Product of Distinct Linear Factors
17:10
Example I: Integration by Partial Fractions
20:33
Case 2: D(x) is a Product of Linear Factors
40:58
Example II: Integration by Partial Fractions
44:41
Integration by Partial Fractions II

42m 57s

Intro
0:00
Case 3: D(x) Contains Irreducible Factors
0:09
Example I: Integration by Partial Fractions
5:19
Example II: Integration by Partial Fractions
16:22
Case 4: D(x) has Repeated Irreducible Quadratic Factors
27:30
Example III: Integration by Partial Fractions
30:19
VII. Differential Equations
Introduction to Differential Equations

46m 37s

Intro
0:00
Introduction to Differential Equations
0:09
Overview
0:10
Differential Equations Involving Derivatives of y(x)
2:08
Differential Equations Involving Derivatives of y(x) and Function of y(x)
3:23
Equations for an Unknown Number
6:28
What are These Differential Equations Saying?
10:30
Verifying that a Function is a Solution of the Differential Equation
13:00
Verifying that a Function is a Solution of the Differential Equation
13:01
Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation
17:20
General Solution
22:00
Particular Solution
24:36
Initial Value Problem
27:42
Example I: Verify that a Family of Functions is a Solution of the Differential Equation
32:24
Example II: For What Values of K Does the Function Satisfy the Differential Equation
36:07
Example III: Verify the Solution and Solve the Initial Value Problem
39:47
Separation of Variables

28m 8s

Intro
0:00
Separation of Variables
0:28
Separation of Variables
0:29
Example I: Solve the Following g Initial Value Problem
8:29
Example II: Solve the Following g Initial Value Problem
13:46
Example III: Find an Equation of the Curve
18:48
Population Growth: The Standard & Logistic Equations

51m 7s

Intro
0:00
Standard Growth Model
0:30
Definition of the Standard/Natural Growth Model
0:31
Initial Conditions
8:00
The General Solution
9:16
Example I: Standard Growth Model
10:45
Logistic Growth Model
18:33
Logistic Growth Model
18:34
Solving the Initial Value Problem
25:21
What Happens When t → ∞
36:42
Example II: Solve the Following g Initial Value Problem
41:50
Relative Growth Rate
46:56
Relative Growth Rate
46:57
Relative Growth Rate Version for the Standard model
49:04
Slope Fields

24m 37s

Intro
0:00
Slope Fields
0:35
Slope Fields
0:36
Graphing the Slope Fields, Part 1
11:12
Graphing the Slope Fields, Part 2
15:37
Graphing the Slope Fields, Part 3
17:25
Steps to Solving Slope Field Problems
20:24
Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y
22:38
VIII. AP Practic Exam
AP Practice Exam: Section 1, Part A No Calculator

45m 29s

Intro
0:00
Exam Link
0:10
Problem #1
1:26
Problem #2
2:52
Problem #3
4:42
Problem #4
7:03
Problem #5
10:01
Problem #6
13:49
Problem #7
15:16
Problem #8
19:06
Problem #9
23:10
Problem #10
28:10
Problem #11
31:30
Problem #12
33:53
Problem #13
37:45
Problem #14
41:17
AP Practice Exam: Section 1, Part A No Calculator, cont.

41m 55s

Intro
0:00
Problem #15
0:22
Problem #16
3:10
Problem #17
5:30
Problem #18
8:03
Problem #19
9:53
Problem #20
14:51
Problem #21
17:30
Problem #22
22:12
Problem #23
25:48
Problem #24
29:57
Problem #25
33:35
Problem #26
35:57
Problem #27
37:57
Problem #28
40:04
AP Practice Exam: Section I, Part B Calculator Allowed

58m 47s

Intro
0:00
Problem #1
1:22
Problem #2
4:55
Problem #3
10:49
Problem #4
13:05
Problem #5
14:54
Problem #6
17:25
Problem #7
18:39
Problem #8
20:27
Problem #9
26:48
Problem #10
28:23
Problem #11
34:03
Problem #12
36:25
Problem #13
39:52
Problem #14
43:12
Problem #15
47:18
Problem #16
50:41
Problem #17
56:38
AP Practice Exam: Section II, Part A Calculator Allowed

25m 40s

Intro
0:00
Problem #1: Part A
1:14
Problem #1: Part B
4:46
Problem #1: Part C
8:00
Problem #2: Part A
12:24
Problem #2: Part B
16:51
Problem #2: Part C
17:17
Problem #3: Part A
18:16
Problem #3: Part B
19:54
Problem #3: Part C
21:44
Problem #3: Part D
22:57
AP Practice Exam: Section II, Part B No Calculator

31m 20s

Intro
0:00
Problem #4: Part A
1:35
Problem #4: Part B
5:54
Problem #4: Part C
8:50
Problem #4: Part D
9:40
Problem #5: Part A
11:26
Problem #5: Part B
13:11
Problem #5: Part C
15:07
Problem #5: Part D
19:57
Problem #6: Part A
22:01
Problem #6: Part B
25:34
Problem #6: Part C
28:54
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Lecture Comments (7)

1 answer

Last reply by: Professor Hovasapian
Fri Aug 18, 2017 3:17 AM

Post by Kevin Wang on August 8, 2017

Hi Professor Hovasapian,

In Example 6, at 35:16, how is it that there isn't a coefficient of 2 in front of the term cos(y)cos(x) ?

1 answer

Last reply by: Professor Hovasapian
Tue Jan 31, 2017 6:46 AM

Post by Efrain Loeza Martinez on January 27, 2017

Lesson : Implicit Differentiation.
Hi professor. In Example V ( 30:00)
¿Why do you cancel the 6?

2 answers

Last reply by: Professor Hovasapian
Fri Nov 18, 2016 8:26 PM

Post by Muhammad Ziad on November 15, 2016

Hi Professor Hovasapian,

In Example 2, why is the derivative of 7y^2, written as 14yy', instead of just 14y.

Thank you!

Implicit Differentiation

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Implicit Differentiation 0:09
    • Implicit Differentiation
  • Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y 12:15
  • Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 14 19:18
  • Example III: Find (dy)/(dx) of x³y² + y³x² = 4x 21:43
  • Example IV: Find (dy)/(dx) of the Following Equation 24:13
  • Example V: Find (dy)/(dx) of 6sin x cos y = 1 29:00
  • Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y 31:02
  • Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x 37:36
  • Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²-y²) 41:03
  • Example IX: Find (d²y)/(dx²) of x² + y² = 25 44:05
  • Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x) 47:48

Transcription: Implicit Differentiation

Hello, welcome back to www.educator.com and welcome back to AP Calculus.0000

Today, we are going to talk about this technique called implicit differentiation.0004

Let us jump right on in.0008

Suppose, we are given the following function.0011

Suppose, we are given x³ + y = 7x, what is dy dx?0019

What is dy dx or y‘, whichever symbol you want to use.0035

For this one, we can solve explicitly for y and just differentiate with we have been doing all this time.0044

By explicitly, I mean just y on one side of the equal sign, and then, everything else in x on the other side of the equal sign.0053

Here we can solve explicitly for y, then differentiate as before.0062

Basically, I just move everything over and I end up with y is equal to 7x -3x³.0091

And then, y’ is 7 -, it is just x³ not 3x³, my apologies.0099

This just becomes -3x².0111

This one is easy to handle.0114

In other words, your function is always going to be given to you in terms of y = this.0116

Sometimes, you have to put it in that form, before you actually take the derivative.0121

What about this, what about x³ + y⁴ = -9xy.0125

How do we handle something like this?0140

Here we cannot solve explicitly for y.0144

We cannot rearrange this equation or manipulate it mathematically such that y = something.0147

Even if we could, the expression might be so complicated.0153

By taking the derivative of it is going to be intractable.0157

It is just not something you want to do.0160

Fortunately, there is a way around this.0162

This equation, it still defines, if a relation between x and y.0165

It is saying that if I x³, I had the y⁴, it is equal to -9xy.0171

There is a relation here between x and y, but the relation is implicit.0175

Explicit means you have one variable on one side of the equality sign.0180

And then, the function whatever it is on the other side, only that variable.0184

Y is a function of x, y = something in x.0189

Here it is implicit, it is implied in this.0192

The idea is that theoretically, you might be able to but how do we find the derivative of it, if we do not have y = something.0195

There is a way of doing so and that is called implicit differentiation, when you write all of these out.0203

Here, we cannot or do not want to express y explicitly, in terms of x.0211

But, the equation, it still defines a relation between x and y, an implicit relation.0240

In other words, the x and y are sort of mixed up in the equation, that is an implicit relation.0269

To find dy dx or y’ with respect to x, we use implicit differentiation.0274

Very important.0294

Here is how you do it.0299

Let me go to blue here.0301

Treat y as a function of x.0303

When you differentiate y, when you take the derivative of y, use the chain rule.0318

It does not really mean much as written.0338

Let us see what actually happens.0339

Use the chain rule then isolate the symbol dy dx.0342

Let us see what this means.0359

Let us start with letters that we had.0362

We had x³ + y⁴ = -9xy.0363

We are going to implicitly differentiate this entire thing, left side and right side.0371

We will start with differentiate everything.0377

You are differentiating with respect to x because that is what we are looking for.0387

We are looking for dy dx.0390

The variable that we are differentiating with respect to is x.0392

It is going to look like this.0396

We are going to take the ddx of x³ + the ddx of y⁴ = the ddx of -9xy.0398

That is all we are saying. We are saying, here we have this equation.0413

What we do to the left side, we do to the right side.0417

If we differentiate the left side, we differentiate the right side.0419

It retains the equality.0422

That is all we are doing.0424

You differentiate everything.0425

The derivative with respect to x³, that is going to be 3x².0427

The derivative with respect to x of y⁴, we treat y as a function of x.0434

Chain rules says, it becomes 4y³ dy dx.0442

That is what the chain rule is.0450

It says this is a global function, there are two things going on.0451

There is the y⁴, there is the power function, and then there is y itself which is a function of x.0455

We are presuming that it is a function of x.0461

It is implicit that it is a function of x.0463

Therefore, we have to write that dy dx, the derivative of -9xy.0465

This -9 is a constant, xy, we use the product rule.0472

It is going to be -9 × this × the derivative of that.0477

It is going to be x × dy dx + y × the derivative of that.0482

Y × 1, the derivative of x with respect to x is 1.0488

We have 3x² + 4y³ dy dx =, I’m going to distribute this, the 9 over this.0497

- 9x dy dx - 9y.0511

I’m going to put all the terms involving dy dx together on one side.0522

It is going to be 4y³ dy dx.0527

I’m going to bring this one over, + 9x dy dx.0537

Over on the other side, I’m going to put everything else.0542

I’m going to move this over.0544

It is going to be -3x² – 9y.0546

I have 4y³ dy dx + 9x × dy dx = this, factor out the dy dx.0552

I get dy dx × 4y³ + 9x = - 3x² -9y, now divide by 4y³ + 9x.0566

In other words, isolate the dy dx.0586

Dy dx = -3x² -9y/ 4y³ + 9x.0589

I have my derivative.0600

The only issue with this derivative is not really an issue.0605

It is just something that you have not seen, is that these derivative actually involves both variables x and y.0607

When a function is given explicitly like y = sin (x), the derivative is only going to involve the variable x.0613

Implicit derivatives, implicit differentiation expresses the derivative in terms of both variables x and y.0621

That is not a problem because in general, we are going to know what x and y are.0627

In the sense that we are going to pick some point, 5 3, -6 9, to put them in to find the derivative.0632

Again, the derivative is two things, it is a slope and it is a rate of change.0640

That is all it is.0644

Having it expressed with a variable itself, the y itself does not really matter.0645

For any value of x and y, this gives the slope at that point.0652

That is it, no big deal.0670

We are just adding that y in there.0673

Notice, dy dx is expressed in terms of both x and y.0679

It is generally going to be true.0703

Again, this is not a problem.0705

We want an expression for dy dx for y’.0709

This implicit differentiation gives us that expression.0713

Nice and straightforward.0718

The best thing to do is just do examples.0720

Once you see the examples, get used to this idea of differentiating y, treating it as of function of x using chain rule on it.0722

Once you see it a couple of times, it will make sense what is going on.0730

Let us jump right on in.0734

We have our first example, xy + 4x - 3x² = 9.0737

Here we want you to find dy dx by both implicit differentiation and solving explicitly for y.0746

Here we want you to do it both ways, to see if we actually get the same answer.0754

Let us see what we have.0758

Let us do it explicitly first.0759

This is going to be explicit.0761

We have our function here.0767

Explicit means we want y = some function of x.0770

Fortunately, this one, we can do.0773

I have the function of xy + 4x - 3x² = 9.0776

That gives me, xy = 9 - 4x + 3x², y = 9 - 4x + 3x²/x.0785

Now we want the derivative, so y’.0804

I’m just going to go ahead and use quotient rule here.0809

This × the derivative of that - that × the derivative of this/ the square.0810

We are going to have x × the derivative of this which is -4 + 6x - that × the derivative of this, -9 - 4x + 3x² × 1/ x².0815

Let me see, this gives me -4x.0840

I’m going to distribute the x, + 6x².0843

This is going to be -9, this is going to be +4x, this is going to be -3x²/ x².0847

-4x and +4x go away.0857

6x² and -3x², that is going to be 3x² - 9/ x².0859

My explicit, in this particular case, was able to separate the x and y, separate the variables.0869

I did my derivative the normal way.0876

Let us do it implicitly and see if we get the same answer.0881

Let us see, where are we?0888

Did I skip anything?0893

No, I did not skip anything.0902

Let me go ahead and do it implicitly.0904

3x² – 9/ x², let me just write that.0907

3x² – 9/ x², this was the derivative that we got.0911

Now let us do it implicitly.0916

I have xy + 4x – 3x² = 9, differentiate everything across the board.0924

This is differentiating everything with respect to x because we are looking for dy dx.0940

This is product rule, it is going to be this × the derivative of that.0946

I’m going to get x × the derivative of y xy‘ + y × the derivative of this + y × 1 + the derivative of 4x is 4.0950

The derivative of this is 6x and the derivative of 9 is 0.0963

I just differentiated right across the board, left side and right side.0969

I'm going to write this as xy’ =, I’m going to move everything over.0975

It is going to be 6x - y – 4.0982

I get y‘ is equal to 6x - y - 4/ x.0989

I found my y‘, my dy dx is 6x - y - 4/ x, implicit.1000

It involves both x and y.1006

Wait a minute, we are supposed to get the same thing.1008

When we did it explicitly, we ended up with 3x² - 9/ x², why are not these the same?1011

They are the same.1021

Let us see what we have got here.1024

We said, remember when we solved explicitly for this function.1028

We got y is equal to 9 - 4x + 3x²/ x.1032

We solved it explicitly.1043

Let us put this expression for y into here, to see what happens.1045

Y‘ is equal to 6x - 9 - 4x + 3x²/ x - 4/ x.1053

Let us simplify this out.1072

I’m going to get a common denominator here.1073

This is going to be 6x².1076

I’m going to distribute -9, that is a -9 + 4x - 3x².1084

The common denominator here is -4x.1098

All of this is over the common denominator x and all of this is /x.1102

4x - 4x, 6x² - 3x² = 3x² – 9/ x².1107

They are the same, the only difference is the implicit differentiation ended up expressing the derivative dy dx, in terms of both x and y.1121

In this particular case, because we had an explicit form, we ended up checking it by putting it in here1131

and realizing that we actually did get that.1137

Again, you are not always going to be able to do that.1139

In this particular case, you were able to.1141

But most implicitly defined relations, you are not going to be able to solve explicitly for one of the variables.1144

You have to leave it in terms of x and y.1149

This is a perfectly good derivative.1152

Example 2, x³ + x² y + 7y² = 14, find dy dx.1160

The biggest problem with calculus is the algebra.1168

Go slow, stay calm, cool and collected, and hopefully everything will work out right.1170

I say this and yet I, myself make mistakes all the time.1176

Hopefully we will keep our fingers crossed and we would not make any algebra mistakes.1179

Let us see how we deal with this one.1183

This is our function and we want to find dy dx.1185

I’m going to work with y’.1192

I do not want to write dy dx over and over again.1193

The derivative of this is 3x², x² y, this is product rule.1197

It is going to be + this × the derivative of that.1205

It is going to be x² × the derivative of y which is just y' + that × the derivative of this.1210

+ y × the derivative of x² is 2x.1220

The derivative of 7y², it is 14y y’.1225

Or we do the chain rule, we take care of the power and then we take dy dx.1231

We write the y'.1237

The derivative of 14 is 0.1239

We are going to go ahead and put things that involve the y or the dy dx together.1244

We have x² y' + 14y y1' =, I’m going to move this and this over to the other side.1251

-3x² -2xy, factor out the y'.1263

Y’ × x² + 14y = -3x² - 2xy, and then divide.1270

I get y’ is equal to -3x² - 2xy all divided by x² + 14y.1282

That is it, that is my derivative, that is dy dx, that is y’.1296

Let us move on to next example.1305

Here we have x³ y² + y³ x² = 4x, find dy dx.1312

Here same thing, this is going to be product rule.1322

I have this × the derivative.1325

I got x³ × the derivative of y² which is 2y y' + y² × the derivative of x³1327

which was 3x² + y³ × the derivative of x² which is 2x + x² × the derivative of y³ which is 3y² dy dx y'.1342

The derivative of 4x is 4, put my prime terms together.1367

I’m going to write this as 2x³ y y' + 3x² y² y’ = 4 - 3x² y².1377

I move this over to that side.1399

I’m going to move this over to that side, - 2xy³, factor out the y'.1401

Y' × 2x³ y + 3x² y² = 4 - 3x² y² - 2xy³, and then divide.1410

Y’ is equal to this whole thing.1428

-3x² y² - 2xy³ divided by 2x³ y + 3x² y².1431

It is exhausting, is not it?1449

It really is, calculus is very exhausting.1451

Y e ⁺x² + 14 = √2 + x² y.1460

Let us dive right on in.1468

Let me go to blue here.1470

This is product rule, this × the derivative of that + that × the derivative of this.1472

We have y × the derivative of this which is e ⁺x² × 2x + e ⁺x² × the derivative of y which is just y'.1477

Just writing y’, dy dx is the derivative, that is what I'm looking for.1495

The derivative of 14 is 0, the derivative of this, this is we know is equal to 2 + x² y ^ ½.1504

The derivative of that is going to be ½ × 2 + x² y⁻¹/2 × the derivative of what is inside which is 0 + this × the derivative of that.1517

X² y’ + this × the derivative of that, x² y’ + y × 2x.1533

Putting all this together, we are going to end up with 2xy e ⁺x² + e ⁺x ² y' =,1539

I’m going to distribute this thing over that and that.1553

This first one is just 0 = ½ x² y' × 2 + x² y⁻¹/2 + this × the other term.1561

½ × 2xy × 2 + x² y⁻¹/2.1582

Let me see, where am I?1606

Now I’m going to go ahead and collect.1609

Let me go ahead and do red.1610

This is something that involves a y' term.1612

This term is what involves a y’ term, I’m going to bring them together.1615

I’m going to get e ⁺x² y’ - ½ x² y' × 2 + x² y⁻¹/2 =, I’m going to put everything else on the other side.1619

I leave this over here, the 2 cancel.1637

I'm left with xy × 2 + x² y⁻¹/2.1640

I bring that over to that side, -2xy e ⁺x².1648

Factor out the y', I get y' × e ⁺x² - ½ x² × 2 + x² y ^-½ = xy × 2 + x² y ^ -½ -2xy e ⁺x².1656

We finally have y’ = xy × 2 + x² y⁻¹/2 - 2xy e ⁺x² all divided by e ⁺x² – ½ 2 + x² y – ½.1692

There you go, that is our final answer.1725

Very complicated looking, but again, once you have x and y, you just plug them in and you solve it.1729

Let us see what we have got.1738

That was example 4.1740

Example 5, 6 sin x cos y = 1.1744

Let me go back to blue here.1748

Let me do it down here.1756

6 ×, this is product rule, sin x × cos y.1761

It is going to be this × the derivative of that + that × the derivative of this.1765

Sin x × the derivative of cos y which is –sin y × y' + cos y × the derivative of sin x which is cos x.1771

The derivative of 1 = 0.1786

We end up with, 6 goes away, we end up with - sin x sin y × y' + cos x cos y = 0.1791

We have -sin x sin y y' = -cos x cos y.1812

Therefore, y' is equal to cos x cos y, I will leave the minus sign, that is fine, divided by -sin x.1832

Sin y, I can go ahead and cancel.1845

I can just leave it as cos x cos y/ sin x sin y.1848

The negatives cancel.1851

Or I can write it as cot x cot y, either one of these is absolutely fine.1852

Hope that makes sense.1862

Example 6, same thing, trigonometric, just more complicated.1866

Product rule, it is going to be a long differential.1877

This × the derivative of that.1884

X² × 2 × cos(y) y'.1886

The derivative of cos² is 2 cos y × the derivative of what y is, y’.1893

Wait, hold on a second, let me do this right.1906

This × the derivative of this.1911

X² × the derivative of cos² y, that is going to be 2 cos y × the derivative of the cos y1913

which is × -sin y × the derivative of y which is y’.1926

There we go, that is better. This × the derivative of that + cos² y × the derivative of x² which is 2x.1931

Now, this one, + y × the derivative of sin x which is cos x + sin x × the derivative of y which is y’.1944

All of this is equal to 2 × this × the derivative of that which is sin x × -sin y y' + cos y × cos x.1958

I do not really need to simplify it, I can just go ahead and jump right on into what it is.1997

That is fine, I will just go ahead and rewrite it.2002

Let me do it in red, I’m making this a little bit simpler.2010

-2x² cos y sin y y’2018

+ 2x cos² y + y cos(x) + sin x y’.2030

Let me make sure I have everything right here.2056

This × the derivative of that, cos x.2068

Wait a minute, this is 2 sin x.2086

Okay, everything looks good.2092

+ sin xy, good, all of that is equal to -2 sin x sin y y’ + cos y cos x.2096

I’m going to bring everything, I’m going back the blue.2118

This term has a y' in it.2123

This term has a y' in it.2128

This term has a y' in it.2130

I’m going to bring all of those over to one side.2132

That is going to be -2x² cos y sin y y' + sin x y' + 2 sin x sin y y' and all of that is going to equal,2136

I’m going to put them this term and this term, I’m going to bring those over that side.2162

= cos y cos x - 2x cos² y – y × cos(x).2166

Let me go back to red.2186

When I factor out the y’ and I divide, I end up with y’ = this thing on the right.2187

Cos y cos(x) – 2x cos² y – y × cos(x) divided by this -2x² cos(y) sin y + sin(x) + 2 sin x sin y.2199

Keeping track of it all, that is it.2233

The fundamental part is differentiating this very carefully.2236

There we are, this is calculus, welcome to calculus.2244

I have an extra page for that too.2251

Great, I’m squeezing everything into one page.2252

Example 7, the √xy = 7 + y² e ⁺x.2257

We just do the same thing that we always do.2264

We know that this is equal to xy¹/2.2268

Let me differentiate.2274

We end up with, it is going to be ½ xy ^-½ × the derivative of what is inside the xy,2275

which is going to be this × the derivative of that xy' + y × the derivative of the x which is 1 = 0 + this × the derivative of that,2297

y² e ⁺x + e ⁺x × the derivative of that which is 2y y'.2310

Let me distribute this part.2319

I’m going to distribute this over that and this over that.2330

I'm going to get ½ x y' × xy ^-½ + ½ y × xy⁻¹/2 = y² e ⁺x + 2y e ⁺x y'.2334

Here is a y’ term, let me go back to blue.2371

Here is y’ term, here is a y’ term.2375

Let me put those together.2378

I have ½ x y’ xy⁻¹/2 – 2y y’ e ⁺x.2380

Let me move this over to that side.2392

It equals y² e ⁺x – ½ y xy⁻¹/2.2395

Let me go back to red.2406

When I factor out the y’, I’m going to get y’ × ½ x × xy⁻¹/2 – 2y e ⁺x = y² e ⁺x – ½ y xy⁻¹/2.2407

I will go ahead and divide.2435

I’m left with y’ = y² e ⁺x – ½ y × xy⁻¹/2 divided by ½ x × xy⁻¹/2 – 2y e ⁺x.2436

There you go.2457

Example number 8, same thing, just equation, any other equation that we have to deal with.2467

Let us go ahead and differentiate this.2476

This is going to be 4 × this thing.2477

The derivative of this thing is going to be 2 × this x² + y² × the derivative of what is inside.2483

That is going to be 2x + 2y y' = 35 × 2x – 2y y.2492

I get 8 ×, I’m going to multiply this.2513

This is a binomial × a binomial.2519

I’m going to get 2x³.2521

This × this is going to be + 2x² y y' + y² × 2x is going to be 2x y² + 2y³ y' = 70x-70y y'.2525

Here I have a y' term, here I have a y' term, here I have a y’ term.2550

Let me go back to red, put a parentheses around that.2556

Let me multiply through, all of these becomes 16, that stays 70.2559

I’m going to go ahead and just multiply, and move things around.2563

On this slide, I'm going to have 16.2568

Let me go to blue.2572

I’m going to have 16x² y y' + 16 y³ y'.2575

I’m going to bring the 70 y’ over + 70y y' = 70x - 16 x³, that is this one.2586

And then, - 16xy².2601

Go back to red, there is my y’.2608

Y’ and y’, factor out, divided by what is left over.2610

I’m going to be left with y' is equal to 70x -16x³ - 16xy² divided by 16x² y + 16y³ + 70 y.2614

There is my derivative.2638

We have x² + y² = 25.2647

This time they want us have to find d² y dx².2649

They want us to find y”, the 2nd derivative.2652

I will just do it in black.2656

X² + y² = 25.2660

The derivative of x² is 2x.2662

The derivative of y² is 2y dy dx, y’ = 0.2665

I have 2y y' = -2x.2674

Therefore, y’ = -x/y, that is my y'.2682

I want y”.2690

Now I take the derivative of this.2694

Again, I differentiate implicitly.2699

Here, this is going to come down.2702

This is going to end up becoming, it is going to be this × the derivative of that - that × the derivative of this/ this².2705

Let me go ahead and just keep my negative sign here.2715

Let me write y” =, the negative sign that is this negative sign.2720

It is going to be this × the derivative of that, y × 1 - this × the derivative of that – x y'/ y².2726

That is equal to distribute the negative sign.2739

We get x y' - y/ y².2741

However, notice the second derivative, not only does it have x and y but it also has the y'.2751

I already know what y' is, I already found the first derivative.2760

Y' is equal to -x/y.2762

I can take this -x/y, stick it into y'.2765

Let me do it down here.2774

This =, y” = x × y' which is - x/y - y/ y²2775

which is equal to -x²/ y - y/ y² = -x² - y²/ y/ y²,2791

which becomes -x² - y²/ y³.2813

When you take the second derivative, the second derivative is actually going to involve the first derivative.2824

But you already found the first derivative, you can put that back in.2830

Something like this, reasonably simple and straightforward.2834

Not going to be so reasonable and straightforward, when you actually do some of the longer problems like we are going to do in a second.2840

You would have to decide the extent to which you want to actually put in.2845

The extent to which you want to simplify it, things like that.2853

As long as you realize that, y” is going to also contain the first derivative and you would already have the 1st derivative.2856

Those are going to be the important parts.2864

Example 10, let us see what we can do.2870

Example 10, let me go ahead and go back to blue.2878

The derivative of sin x is the cos(x).2886

The derivative of cos y =, it is negative, + -sin y y1'.2890

The derivative of sin 2x is cos 2x × the derivative of 2 which is 2, it is 2 × cos(2x).2903

Let us solve, -sin y y’ is equal to 2 cos 2x – cos x.2916

Therefore, we get y' is equal to 2 cos 2x - cos x/ -sin y.2928

I’m going to go ahead and take this negative sign, bring it up top.2942

Flip these two and I’m going to write this as cos(x) -2 cos(2x)/ sin y.2945

I hope that made sense.2958

A negative sign actually can go top or bottom.2958

It does not really matter.2960

I went ahead and brought it up here.2961

When I distributed over this, the negative of this and this, this one becomes positive, I put it first.2964

This one is negative, I put it second.2969

That is all I have done.2971

There we go, we have y’.2973

Now, y”.2977

Y”, let me do this one in red.2982

Y”, I'm actually going to be differentiating this one implicitly.2988

It is going to be this × the derivative of that - that × the derivative of this/ this².2995

That = sin y, this × the derivative of this.3001

The derivative of cos x is -sin x.3010

The derivative of this cos 2x is going to be -2 sin 2x.3017

-2 × 2 is 4, it is going to be +, it is going to be +4, × sin(2x).3025

That is this × the derivative of that - this cos x - 2 cos 2x × the derivative of this which is cos y y’/ sin² y.3032

Let us take a look at this, before we do anything.3059

It involves sin x, it involves sin y, or cos x and cos y.3065

It looks like the only y' term that we have is here.3070

That is this one.3075

If you were going to simplify it, you do not have to.3077

If you actually have to, for y', this is y'.3080

You take this expression, you put it into here.3087

And then, you simplify the expression as much as possible.3091

I’m not going to go ahead and do this.3094

For this particular problem, something that is going to end up looking like this, I would just leave it alone.3096

If your teacher is going to have you do this, where you actually have to substitute in,3103

only express things in terms of x and y, then you have to do something simple.3106

Otherwise, you will be here for 4 days doing this.3110

But for something like this, you are going to deal with it at some point in your career.3113

You have y', you have y”, if you need to, you go ahead and put it in.3119

Again, most of the time, when you are working,3123

you are going to be working with mathematical software so it is going to be doing this for you.3126

That is it, straight differentiation.3130

You differentiate the first one implicitly.3134

You get this, you differentiate the second one implicitly.3135

It is going to get progressively more complicated, simply by virtue of the fact that this happens to be a quotient rule.3139

I hope that that made sense.3147

Thank you so much for joining us here at www.educator.com.3148

We will see you next time, bye.3150

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