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College Calculus: Level I The Power Rule

Section 3: Derivatives, part 1: Lecture 2 | 26:01 min

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Lecture Comments (13)

	1 answer Last reply by: Dr. Jennifer Switkes Mon Nov 12, 2018 6:57 PM Post by Samatar Farah on October 30, 2018 The audio files for the additional examples do not exist.
	0 answers Post by Vance Bower on February 22, 2016 On the practice questions, there is a derivative that is "Y=x^1/3" or y=x tot eh one third. They say the answer is 1/3x^-1/3. Why isn't that 1/3 x^-2/3, if f(x^n)=nx^n-1?
	1 answer Last reply by: Mohamed Al Mohannadi Sat Sep 10, 2016 10:34 AM Post by Maryam Ahmad on October 21, 2015 I swear I just wana say that People like me; students like me who can not afford 50 or 60 bucks for an hour and are so keen to learn and do good in exams so we can have better grade for class...are struggling so HARD!! Seriously My Heart aces to say that the system of Education in United States is not what I hoped for when I came here.. Professors simply just DONT CARE!! (talking about Uni Level tho) .. All they care is to be ahead of syllabus.. and dn't get me wrong I am PAYING for this but this does not HURT because the explanation was so beautiful that I actually understood!! and it is sooo SAD when you pay 25,000$ an year un UNI but its not worth it because there is no such thing TEACHING happening and there is no such thing LEARNING happening...!! A humble THANK YOU from a struggling Immigrant Student (I really had a lot bottled up on my chest)
	0 answers Post by Andrew Demidenko on June 7, 2015 http://en.wikipedia.org/wiki/Power_rule
	0 answers Post by john doe on December 18, 2014 Very smooth teaching - no stuttering and pauses so easy to follow the train of thought. Very well done!!!!
	0 answers Post by Eric Nunez on March 20, 2013 I just have to say that in Example 3 problem (i) your detailed explanation of (7x) and the constant 3 blew my mind. When you forced it into the structure of the power rule a switch flipped for me. Thank you so much for your ultra detailed explanation.
	2 answers Last reply by: abbas esmailzadeh Sat Dec 3, 2011 5:43 PM Post by abbas esmailzadeh on December 3, 2011 when i logoff and logon how can i trace the last lecture i takeoff from
	1 answer Last reply by: Pamela Larson Fri Jan 21, 2011 11:49 AM Post by Pamela Larson on January 21, 2011 Derivatives, part 1; the power rule, Example 3 is not working at approx. 14:05...... Can you please fix this, Thanks

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The Power Rule

Check whether your instructor wants you to know the proof of the Power Rule of Differentiation.
Practice carrying out the Power Rule on problems involving negative exponents and fractional exponents.
The derivative of a sum (or difference) is the sum (or difference) of the individual derivatives.
A constant factor can be factored out in front of a derivative.

The Power Rule

Find the derivative of y = 2x³ + 5x² + 7x + 11

dy = [d/dx] (2x³ + 5x² + 7x + 11)
= 2[d/dx] x³ + 5 [d/dx] x² + 7 [d/dx] x
= 2(3x²) + 5(2x) + 7(1) =

6x² + 10x + 7

Find the derivative of y = x^[1/3]

dy = [d/dx] x^[1/3]

[1/3] x^−[1/3]

Find the derivative of y = (x + 4)²

dy = [d/dx] (x + 4)²
= [d/dx] (x² + 8x + 16)=

2x + 8

Find the derivative of y = √2 x²

dy = [d/dx] √2 x²
= √2 [d/dx] x² =

2 √2 x

Find the derivative of y = 2 √x

dy = [d/dx]2 √x
= 2 [d/dx] √x
= 2 [d/dx] x^[1/2]
= 2 [1/2] x^−[1/2]
= x^−[1/2] =

[1/(√x)]

Find the derivative of y = [(x³)/(x^[1/3])]

dy = [d/dx] [(x³)/(x^[1/3])]
= [d/dx] x³ x^−[1/3]
= [d/dx] x^{(3 − [1/3])}
= [d/dx] x^[8/3]

[8/3] x^[5/3]

Find the derivative of y = (√x + 1)²

dy = [d/dx] (√x + 1)²
= [d/dx] (x + 2√x + 1)

= 1 + x^−[1/2]

Find the derivative of y = x − [(x³)/3!] + [(x⁵)/5!]

dy = [d/dx] (x − [(x³)/3!] + [(x⁵)/5!])
= 1 − [(3x²)/3!] + [(5x⁴)/5!]

1 − [(x²)/2!] + [(x⁴)/4!]
NOTE: y in this problem is a truncated version of the series representation of sin(x).

Find the derivative of y = 1 − [(x²)/2!] + [(x⁴)/4!] − [(x⁶)/6!]

dy = [d/dx] (1 − [(x²)/2!] + [(x⁴)/4!] − [(x⁶)/6!])
= 0 − [2x/2!] + [(4x³)/4!] − [(6x⁵)/6!]

−x + [(x³)/3!] − [(x⁵)/5!]
NOTE: y in this problem is a truncated version of the series representation of cos(x).

Find the derivative of y = 1 + x + [(x²)/2!] + [(x³)/3!] + [(x⁴)/4!]

dy = [d/dx] (1 + x + [(x²)/2!] + [(x³)/3!] + [(x⁴)/4!])

1 + x + [(x²)/2!] + [(x³)/3!]
NOTE: y in this problem is a truncated version of the series representation of e^x.

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Answer

The Power Rule

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

Power Rule of Differentiation

Lecture Example 1

Lecture Example 2

Lecture Example 3

Additional Example 4

Additional Example 5

Intro 0:00
Power Rule of Differentiation 0:14

Power Rule with Constant
Sum/Difference

Lecture Example 1 1:59
Lecture Example 2 6:48
Lecture Example 3 11:22
Additional Example 4
Additional Example 5

College Calculus 1 Online Course

Section 1: Overview of Functions
	Review of Functions	26:29
	Compositions of Functions	12:29
Section 2: Limits
	Average and Instantaneous Rates of Change	20:59
	Limit Investigations	22:37
	Algebraic Evaluation of Limits	28:19
	Formal Definition of a Limit	23:39
	Continuity and the Intermediate Value Theorem	19:09
Section 3: Derivatives, part 1
	Limit Definition of the Derivative	22:52
	The Power Rule	26:01
	The Product Rule	14:54
	The Quotient Rule	19:17
	Applications of Rates of Change	17:43
	Trigonometric Derivatives	26:58
	The Chain Rule	23:47
	Inverse Trigonometric Functions	27:05
	Equation of a Tangent Line	15:52
Section 4: Derivatives, part 2
	Implicit Differentiation	30:05
	Higher Derivatives	13:16
	Logarithmic and Exponential Function Derivatives	17:42
	Hyperbolic Trigonometric Function Derivatives	14:30
	Related Rates	29:05
	Linear Approximation	23:52
Section 5: Application of Derivatives
	Absolute Minima and Maxima	18:57
	Mean Value Theorem and Rolle's Theorem	20:00
	First Derivative Test, Second Derivative Test	27:11
	L'Hopital's Rule	23:09
	Curve Sketching	40:16
	Applied Optimization	25:37
	Newton's Method	25:13
Section 6: Integrals
	Approximating Areas and Distances	36:50
	Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus	22:02
	Substitution Method for Integration	23:19
Section 7: Application of Integrals, part 1
	Area Between Curves	19:59
	Volume by Method of Disks and Washers	24:22
	Volume by Method of Cylindrical Shells	30:29
	Average Value of a Function	16:31
Section 8: Extra
	Graphs of f, f', f''	23:58
	Slope Fields for Differential Equations	18:32
	Separable Differential Equations	17:04

Related Books

	Calculus by Larson, 9th Edition Authors: Ron Larson, Bruce H. Edwards ISBN: 0547167024 Publisher: Brooks Cole Year: 2009 The book is filled with examples and detailed explanations. Capstone exercises are also included for each section and synthesize the main concepts into a single example.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 1 Author: Ron Larson ISBN: 547213093 Publisher: Brooks Cole Year: 2008 This solutions manual includes worked out solutions to every odd-numbered exercise in Calculus of a Single Variable, 9th edition.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 2 Authors: Ron Larson, Bruce H. Edwards ISBN: 547213107 Publisher: Brooks Cole Year: 2009 This solutions manual includes worked out solutions to every odd-numbered exercise in Multivariable Calculus, 9th edition.

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Professor Switkes

The Power Rule

Slide Duration:

Table of Contents

Section 1: Overview of Functions

Review of Functions

26m 29s

Intro