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Higher Derivatives

Slide Duration:

Section 1: Overview of Functions
Review of Functions

26m 29s

Intro
0:00
What is a Function
0:10
Domain and Range
0:21
Vertical Line Test
0:31
Example: Vertical Line Test
0:47
Function Examples
1:57
Example: Squared
2:10
Example: Natural Log
2:41
Example: Exponential
3:21
Example: Not Function
3:54
Odd and Even Functions
4:39
Example: Even Function
5:10
Example: Odd Function
5:53
Odd and Even Examples
6:48
Odd Function
6:55
Even Function
8:43
Increasing and Decreasing Functions
10:15
Example: Increasing
10:42
Example: Decreasing
10:55
Increasing and Decreasing Examples
11:41
Example: Increasing
11:48
Example: Decreasing
12:33
Types of Functions
13:32
Polynomials
13:45
Powers
14:06
Trigonometric
14:34
Rational
14:50
Exponential
15:13
Logarithmic
15:29
Lecture Example 1
15:55
Lecture Example 2
17:51
-1
-2
Compositions of Functions

12m 29s

Intro
0:00
Compositions
0:09
Alternative Notation
0:32
Three Functions
0:47
Lecture Example 1
1:19
Lecture Example 2
3:25
Lecture Example 3
6:45
-1
-2
Section 2: Limits
Average and Instantaneous Rates of Change

20m 59s

Intro
0:00
Rates of Change
0:11
Average Rate of Change
0:21
Instantaneous Rate of Change
0:33
Slope of the Secant Line
0:46
Slope of the Tangent Line
1:00
Lecture Example 1
1:14
Lecture Example 2
6:36
Lecture Example 3
11:30
-1
-2
Limit Investigations

22m 37s

Intro
0:00
What is a Limit?
0:10
Lecture Example 1
0:56
Lecture Example 2
5:28
Lecture Example 3
9:27
-1
-2
Algebraic Evaluation of Limits

28m 19s

Intro
0:00
Evaluating Limits
0:09
Lecture Example 1
1:06
Lecture Example 2
5:16
Lecture Example 3
8:15
Lecture Example 4
12:58
-1
-2
Formal Definition of a Limit

23m 39s

Intro
0:00
Formal Definition
0:13
Template
0:55
Epsilon and Delta
1:24
Lecture Example 1
1:40
Lecture Example 2
9:20
-1
-2
Continuity and the Intermediate Value Theorem

19m 9s

Intro
0:00
Continuity
0:13
Continuous
0:16
Discontinuous
0:37
Intermediate Value Theorem
0:52
Example
1:22
Lecture Example 1
2:58
Lecture Example 2
9:02
-1
-2
Section 3: Derivatives, part 1
Limit Definition of the Derivative

22m 52s

Intro
0:00
Limit Definition of the Derivative
0:11
Three Versions
0:13
Lecture Example 1
1:02
Lecture Example 2
4:33
Lecture Example 3
6:49
Lecture Example 4
10:11
-1
-2
The Power Rule

26m 1s

Intro
0:00
Power Rule of Differentiation
0:14
Power Rule with Constant
0:41
Sum/Difference
1:15
Lecture Example 1
1:59
Lecture Example 2
6:48
Lecture Example 3
11:22
-1
-2
The Product Rule

14m 54s

Statement of the Product Rule
0:08
Lecture Example 1
0:41
Lecture Example 2
2:27
Lecture Example 3
5:03
-1
-2
The Quotient Rule

19m 17s

Intro
0:00
Statement of the Quotient Rule
0:07
Carrying out the Differentiation
0:23
Quotient Rule in Words
1:00
Lecture Example 1
1:19
Lecture Example 2
4:23
Lecture Example 3
8:00
-1
-2
Applications of Rates of Change

17m 43s

Intro
0:00
Rates of Change
0:11
Lecture Example 1
0:44
Lecture Example 2
5:16
Lecture Example 3
7:38
-1
-2
Trigonometric Derivatives

26m 58s

Intro
0:00
Six Basic Trigonometric Functions
0:11
Patterns
0:47
Lecture Example 1
1:18
Lecture Example 2
7:38
Lecture Example 3
12:15
Lecture Example 4
14:25
-1
-2
The Chain Rule

23m 47s

Intro
0:00
Statement of the Chain Rule
0:09
Chain Rule for Three Functions
0:27
Lecture Example 1
1:00
Lecture Example 2
4:34
Lecture Example 3
7:23
-1
-2
Inverse Trigonometric Functions

27m 5s

Intro
0:00
Six Basic Inverse Trigonometric Functions
0:10
Lecture Example 1
1:11
Lecture Example 2
8:53
Lecture Example 3
12:37
-1
-2
Equation of a Tangent Line

15m 52s

Intro
0:00
Point Slope Form
0:10
Lecture Example 1
0:47
Lecture Example 2
3:15
Lecture Example 3
6:10
-1
-2
Section 4: Derivatives, part 2
Implicit Differentiation

30m 5s

Intro
0:00
Purpose
0:09
Implicit Function
0:20
Lecture Example 1
0:32
Lecture Example 2
7:14
Lecture Example 3
11:22
Lecture Example 4
16:43
-1
-2
Higher Derivatives

13m 16s

Intro
0:00
Notation
0:08
First Type
0:19
Second Type
0:54
Lecture Example 1
1:41
Lecture Example 2
3:15
Lecture Example 3
4:57
-1
-2
Logarithmic and Exponential Function Derivatives

17m 42s

Intro
0:00
Essential Equations
0:12
Lecture Example 1
1:34
Lecture Example 2
2:48
Lecture Example 3
5:54
-1
-2
Hyperbolic Trigonometric Function Derivatives

14m 30s

Intro
0:00
Essential Equations
0:15
Six Basic Hyperbolic Trigc Functions
0:32
Six Basic Inverse Hyperbolic Trig Functions
1:21
Lecture Example 1
1:48
Lecture Example 2
3:45
Lecture Example 3
7:09
-1
-2
Related Rates

29m 5s

Intro
0:00
What Are Related Rates?
0:08
Lecture Example 1
0:35
Lecture Example 2
5:25
Lecture Example 3
11:54
-1
-2
Linear Approximation

23m 52s

Intro
0:00
Essential Equations
0:09
Linear Approximation (Tangent Line)
0:18
Example: Graph
1:18
Differential (df)
2:06
Delta F
5:10
Lecture Example 1
6:38
Lecture Example 2
11:53
Lecture Example 3
15:54
-1
-2
Section 5: Application of Derivatives
Absolute Minima and Maxima

18m 57s

Intro
0:00
Minimums and Maximums
0:09
Absolute Minima and Maxima (Extrema)
0:53
Critical Points
1:25
Lecture Example 1
2:58
Lecture Example 2
6:57
Lecture Example 3
10:02
-1
-2
Mean Value Theorem and Rolle's Theorem

20m

Intro
0:00
Theorems
0:09
Mean Value Theorem
0:13
Graphical Explanation
0:36
Rolle's Theorem
2:06
Graphical Explanation
2:28
Lecture Example 1
3:36
Lecture Example 2
6:33
Lecture Example 3
9:32
-1
-2
First Derivative Test, Second Derivative Test

27m 11s

Intro
0:00
Local Minimum and Local Maximum
0:14
Example
1:01
First and Second Derivative Test
1:26
First Derivative Test
1:36
Example
2:00
Second Derivative Test (Concavity)
2:58
Example: Concave Down
3:15
Example: Concave Up
3:54
Inconclusive
4:19
Lecture Example 1
5:23
Lecture Example 2
12:03
Lecture Example 3
15:54
-1
-2
L'Hopital's Rule

23m 9s

Intro
0:00
Using L'Hopital's Rule
0:09
Informal Definition
0:34
Lecture Example 1
1:27
Lecture Example 2
4:00
Lecture Example 3
5:40
Lecture Example 4
9:38
-1
-2
Curve Sketching

40m 16s

Intro
0:00
Collecting Information
0:15
Domain and Range
0:17
Intercepts
0:21
Symmetry Properties (Even/Odd/Periodic)
0:33
Asymptotes (Vertical/Horizontal/Slant)
0:45
Critical Points
1:15
Increasing/Decreasing Intervals
1:24
Inflection Points
1:38
Concave Up/Down
1:52
Maxima/Minima
2:03
Lecture Example 1
2:58
Lecture Example 2
10:52
Lecture Example 3
17:55
-1
-2
Applied Optimization

25m 37s

Intro
0:00
Real World Problems
0:08
Sketch
0:11
Interval
0:20
Rewrite in One Variable
0:26
Maximum or Minimum
0:34
Critical Points
0:42
Optimal Result
0:52
Lecture Example 1
1:05
Lecture Example 2
6:12
Lecture Example 3
13:31
-1
-2
Newton's Method

25m 13s

Intro
0:00
Approximating Using Newton's Method
0:10
Good Guesses for Convergence
0:32
Lecture Example 1
0:49
Lecture Example 2
4:21
Lecture Example 3
7:59
-1
-2
Section 6: Integrals
Approximating Areas and Distances

36m 50s

Intro
0:00
Three Approximations
0:12
Right Endpoint, Left Endpoint, Midpoint
0:22
Formulas
1:05
Velocity and Distance
1:35
Lecture Example 1
2:28
Lecture Example 2
12:10
Lecture Example 3
19:43
-1
-2
Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus

22m 2s

Intro
0:00
Important Equations
0:22
Riemann Sum
0:28
Integral
1:58
Integrand
2:35
Limits of Integration (Upper Limit, Lower Limit)
2:43
Other Equations
3:05
Fundamental Theorem of Calculus
4:00
Lecture Example 1
5:04
Lecture Example 2
10:43
Lecture Example 3
13:52
-1
-2
Substitution Method for Integration

23m 19s

Intro
0:00
U-Substitution
0:13
Important Equations
0:30
Purpose
0:36
Lecture Example 1
1:30
Lecture Example 2
6:17
Lecture Example 3
9:00
Lecture Example 4
11:24
-1
-2
Section 7: Application of Integrals, part 1
Area Between Curves

19m 59s

Intro
0:00
Area Between Two Curves
0:12
Graphic Description
0:34
Lecture Example 1
1:44
Lecture Example 2
5:39
Lecture Example 3
8:45
-1
-2
Volume by Method of Disks and Washers

24m 22s

Intro
0:00
Important Equations
0:16
Equation 1: Rotation about x-axis (disks)
0:27
Equation 2: Two curves about x-axis (washers)
3:38
5:31
Lecture Example 1
6:05
Lecture Example 2
8:28
Lecture Example 3
11:55
-1
-2
Volume by Method of Cylindrical Shells

30m 29s

Intro
0:00
Important Equations
0:50
1:04
Equation 2: Rotation about y-axis (2 curves)
7:34
8:15
Lecture Example 1
8:57
Lecture Example 2
14:26
Lecture Example 3
18:15
-1
-2
Average Value of a Function

16m 31s

Intro
0:00
Important Equations
0:11
Origin of Formula
0:34
Lecture Example 1
2:51
Lecture Example 2
5:30
Lecture Example 3
8:13
-1
-2
Section 8: Extra
Graphs of f, f', f''

23m 58s

Intro
0:00
Slope Function of f(x)
0:41
Slope is Zero
0:53
Slope is Positive
1:03
Slope is Negative
1:13
Slope Function of f'(x)
1:31
Slope is Zero
1:42
Slope is Positive
1:48
Slope is Negative
1:54
Lecture Example 1
2:23
Lecture Example 2
8:06
Lecture Example 3
12:36
-1
-2
Slope Fields for Differential Equations

18m 32s

Intro
0:00
Things to Remember
0:13
Graphic Description
0:42
Lecture Example 1
1:44
Lecture Example 2
6:59
Lecture Example 3
9:46
-1
-2
Separable Differential Equations

17m 4s

Intro
0:00
Differential Equations
0:10
Focus on Exponential Growth/Decay
0:27
Separating Variables
0:47
Lecture Example 1
1:35
Lecture Example 2
6:41
Lecture Example 3
9:36
-1
-2

• ## Related Books

 0 answersPost by Amina Kalla on February 16, 2013how do we find the nth derivatives of Sin^2(x)

### Higher Derivatives

• This topic simply involves repeatedly differentiating!
• Learn the two different types of notation for higher derivatives – both are used a lot!
• Some problems will ask you to find a formula for the nth derivative – in this case, differentiate enough times to see the pattern and then build a formula describing the pattern.
• Some problems will ask you to find, for example, the 100th derivative – in this case, again differentiate enough times to see the pattern and then figure out the result!

### Higher Derivatives

Find the second derivative of f(t) = 3t2 + 11t + 17
• f′(t) = [d/dt] (3t2 + 11t + 17)
• = 6t + 11
• f"(t) = [d/dt] (6t + 11)
6
Find the third derivative of f(t) = 3t2 + 11t + 17
• f"′(t) = [d/dt] f"(t)
• = [d/dt] (6)
0
Find the second derivative of f(x) = tan(3x)
• f′(x) = [d/dx] tan(3x)
• = sec2(3x) [d/dx] (3x)
• = 3 sec2(3x)
• f"(x) = [d/dx] 3 sec2(3x)
• u = sec(3x), u′ = sec(3x)tan(3x) [d/dx] 3x = 3 sec(3x)tan(3x)
• f"(x) = 3 [d/dx] u2
• = 6u u′
• = 6 sec(3x) (3 sec(3x)tan(3x))
18 sec2(3x) tan(3x)
Find the second derivative of f(x) = x − [(x3)/6] + [(x5)/120]
• f′(x) = [d/dx] (x − [(x3)/6] + [(x5)/120])
• = 1 − [(3x2)/6] + [(5x4)/120]
• = 1 − [(x2)/2] + [(x4)/24]
• f"(x) = [d/dx] (1 − [(x2)/2] + [(x4)/24])
• = 0 − [2x/2] + [(4x3)/24]
−x + [(x3)/6]
f(x) here might look familiar. It is a partial series representation of sin(x)
Find the second derivative of f(t) = t2 sin(5t)
• f′(x) = [d/dt] (t2 sin(5t))
• = t2 [d/dt] sin(5t) + sin(5t) [d/dx] t2
• = t2 (5cos(5t)) + sin(5t) (2t)
• = 5 t2 cos(5t) + 2t sin(5t)
• f"(x) = [d/dt] (5 t2 cos(5t) + 2t sin(5t))
• = 5t2 [d/dt] cos(5t) + 5cos(5t) [d/dt] t2 + 2t [d/dt] sin(5t) + 2 sin(5t) [d/dt] t
• = −25t2sin(5t) + 10t cos(5t) + 10t cos(5t) + 2sin(5t)
−25t2sin(5t) + 20t cos(5t) + 2sin(5t)
Find the second derivative of x(t) = 5 cos(2t − [(π)/4])
• x′(t) = [d/dt] (5 cos(2t − [(π)/4]) )
• = −5 sin(2t − [(π)/4]) [d/dt] (2t − [(π)/4])
• = −10 sin(2t − [(π)/4])
• x"(t) = [d/dt] (−10 sin(2t − [(π)/4]))
• = −10 cos(2t − [(π)/4]) [d/dt] (2t − [(π)/4]) =
• x(t) in this problem represents an example of simple harmonic motion, which can be used to describe the motion of things like springs
−20 cos(2t − [(π)/4])
Find the second derivative of f(x) = [1/(x2)]
• f(x) = [1/(x2)]
• = x−2
• f′(x) = [d/dx] x−2
• = −2x−3
• f"(x) = [d/dx] (−2x−3)
• = −2 [d/dx] x−3
• = −2(−3x−4)
• = 6x−4 =
[6/(x4)]
Find the second derivative of f(x) = √{x3 + 5x}
• f′(x) = [d/dx] √{x3 + 5x}
• u = x3 + 5x, u′ = 3x2 + 5, u" = 6x
• f′(x) = [d/dx] u[1/2]
• = [1/2] u−[1/2] u′
• f"(x) = [d/dx] [1/2] u−[1/2] u′
• = [1/2] ( u−[1/2] [d/dx] u′+ u′[d/dx] (u−[1/2]))
• = [1/2] ( u−[1/2] u" + u′(−[1/2] u−[3/2] u′))
• = [1/2] ( u−[1/2] u" − [1/2] (u′)2 u−[3/2])
[1/2] ( (x3 + 5x)−[1/2] (6x) − [1/2] (3x2 + 5)2 (x3 + 5x)−[3/2])
Find the 2nd, 4th, 6th, and 8th derivatives of y = sin(x)
• Here's another way to write higher order derivatives
• [d/dx] [d/dx]z = [(d2)/(dx2)] z (2nd derivative of z with respect to x)
• [(d3)/(dx3)] (f(x) equivalent to f"′(x))
• [(d9 y)/(dx9)] (9th derivative of y with respect to x)
y = sin(x)
dy = cos(x)
[d/dy]2 = −sin(x)
[d/dy]3 = −cos(x)
[d/dy]4 = sin(x)
[d/dy]5 = cos(x)
[d/dy]6 = −sin(x)
[d/dy]7 = −cos(x)
[d/dy]8 = sin(x)
Find the 2nd, 4th, 6th, and 8th derivatives of y = cos(2x)
y = cos(2x)
dy = −2sin(2x)
[d/dy]2 = −4cos(2x)
[d/dy]3 = 8sin(2x)
[d/dy]4 = 16cos(2x) = 24 cos(2x)
[d/dy]6 = −(26)cos(2x)
[d/dy]8 = (28)cos(2x)

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Higher Derivatives

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• Intro 0:00
• Notation 0:08
• First Type
• Second Type
• Lecture Example 1 1:41
• Lecture Example 2 3:15
• Lecture Example 3 4:57

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