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Calculus AB Higher Order Derivatives

Section 3: Derivatives: Lecture 8 | 15:00 min

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Lecture Comments (6)

	0 answers Post by Quinn Hollister on May 23, 2015 I believe 2nd derivative is acceleration, and first derivative is the velocity function
	0 answers Post by Nolan Zhang on April 10, 2015 example 3 was quite long, you should probably use another one
	0 answers Post by Quazi Niloy on October 13, 2014 On example 3, wouldn't it have been simple to take sin(t)/t^3, and then just multiply it where you have sin(t)*(t^-3) and then apply either the product or the chain rule?
	2 answers Last reply by: John Zhu Tue Aug 13, 2013 8:36 PM Post by Vanessa Munoz on June 12, 2013 I don't understand in 10:17, how you can replace with zero if it makes de denominator zero..wouldn't that be undefined?

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Higher Order Derivatives

Different order derivatives have different practical meanings
To avoid confusion: treat each level of derivative as brand new derivative
Higher order derivatives are easier to solve because of eliminated terms

Higher Order 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)

Find the third derivative of f(t) = 3t² + 11t + 17

f"′(t) = [d/dt] f"(t)
= [d/dt] (6)

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)

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Answer

Higher Order Derivatives

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

Types of Derivatives: Higher Order Derivatives

Example 2: Find 3rd Derivative

Example 3: Acceleration

Example 4

Example 5: 2nd Derivative

Intro 0:00
Types of Derivatives: Higher Order Derivatives 0:07

1st Derivative / F Prime
2nd Derivative
3rd Derivative
Example 1

Example 2: Find 3rd Derivative 3:13
Example 3: Acceleration 4:25
Example 4 10:20
Example 5: 2nd Derivative 12:11

Mathematics: AP Calculus AB

Section 1: Functions
	Definitions & Properties of Functions	11:26
	Graphing	13:58
	Inverse Functions	6:47
	Polynomial Functions	5:04
	Trigonometric Functions	6:45
	Inverse Trigonometric Functions	5:58
	Trigonometric Identities	17:42
	Exponential Functions	5:53
	Logarithmic Functions	7:08
	Rational Functions	15:36
	Conic Sections	14:58
Section 2: Limits and Continuity
	Limit Definition & Properties	7:15
	Solving Limits with Algebra	8:01
	Rational Limit Rules	3:16
	One Sided Limits	9:57
	Special Trigonometric Limits	8:28
	Limits & Continuity	10:14
	Limits: Multiple Choice Practice	6:16
Section 3: Derivatives
	Derivative Definition & Properties	4:11
	Basic Rules of Differentiation	7:07
	Power Rule	7:14
	Trigonometric Rules	7:53
	Product Rule	11:11
	Quotient Rule	16:50
	Chain Rule	19:48
	Higher Order Derivatives	15:00
	Derivatives of Exponential Functions	13:14
	Derivatives of Logarithmic Functions	11:30
	Derivatives of Inverse Trigonometric Functions	16:54
	Implicit Differentiation	16:53
	Multiple Choice Practice: Derivatives	11:07
Section 4: Applications of Derivatives
	Tangent & Normal Lines	22:36
	Position Velocity & Acceleration	18:42
	Related Rates	26:22
	Minimum & Maximum	12:22
	Concavity	11:43
	Rolles Theorem	8:28
	Mean Value Theorem	9:39
	Differentials	12:25
	Applications of Derivatives: Multiple Choice Practice	13:21
	Applications of Derivatives: Free Response Practice	10:22
Section 5: Integrals
	Definition of Integrals	1:08
	Integrals of Power Rule	8:50
	Integrals Basic Rules of Integration	9:43
	Trigonometric Rules of Integrals	8:58
	Chain Rule	13:59
	Integrals of Exponential Functions	12:52
	Integrals of Natural Logarithmic Functions	13:00
	Integrals of Inverse Trigonometric Functions	8:29
	Integrals: Multiple Choice Practice	15:37
Section 6: Applications of Integrals
	Fundamental Theorem of Calculus	15:55
	Area Under A Curve	18:34
	Reimann Sums	10:35
	Trapezoid Rule	12:46
	Mean Value Theorem	11:22
	Second Fundamental Theorem of Calculus	4:44
	Area Between Curves	16:39
	Revolving Solids Washer Disk Methods	21:09
	Revolving Solids Cylindrical Shells Method	26:46
	Revolving Solids Known Cross Sections	27:41
	Differential Equations Eulers Method	17:54
	Differential Equations Slope Fields	16:30
	Application of Integrals: Multiple Choice Practice	14:19
	Application of Integrals: Free Response Practice	9:14
Section 7: Sample AP Test
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 1	17:50
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 2	17:32
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 3	22:14
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 4	19:35
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 5	25:43
	AP Calculus AB Practice Test: Section 2: Free Response Part 1	16:50
	AP Calculus AB Practice Test: Section 2: Free Response Part 2	21:36
	AP Calculus AB Practice Test: Section 2: Free Response Part 3	13:39

Related Books

	Cracking the AP Calculus AB Exam, 2015 Edition Author: Princeton Review ISBN: 804124809 Publisher: Princeton Review Year: 2014 The book includes 3 full length practice tests with detailed explanations, a review of all the key concepts, and targeted strateeies to ace th exam for your highest score.
	Cliffs AP: Calculus AB & BC, 3rd Edition Authors: Dale W Johnson, Kerry J King ISBN: 764586831 Publisher: Cliffs Notes Year: 2001 The book has a topic-by topic breakdown and lots of problem approach suggestions for both free response and multiple choice calculus questions. The book also includes two full length tests.

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John Zhu

Higher Order Derivatives

Slide Duration:

Table of Contents

Section 1: Functions

Definitions & Properties of Functions

11m 26s

Intro