For more information, please see full course syllabus of College Calculus: Level I

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For more information, please see full course syllabus of College Calculus: Level I

For more information, please see full course syllabus of College Calculus: Level I

### 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 = 2x

^{3}+ 5x^{2}+ 7x + 11- dy = [d/dx] (2x
^{3}+ 5x^{2}+ 7x + 11) - = 2[d/dx] x
^{3}+ 5 [d/dx] x^{2}+ 7 [d/dx] x - = 2(3x
^{2}) + 5(2x) + 7(1) =

6x

^{2}+ 10x + 7Find 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] (x
^{2}+ 8x + 16)=

2x + 8

Find the derivative of y = √2 x

^{2}- dy = [d/dx] √2 x
^{2} - = √2 [d/dx] x
^{2}=

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 = [(x

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

^{3})/3!] + [(x^{5})/5!]- dy = [d/dx] (x − [(x
^{3})/3!] + [(x^{5})/5!]) - = 1 − [(3x
^{2})/3!] + [(5x^{4})/5!]

1 − [(x

NOTE: y in this problem is a truncated version of the series representation of sin(x).

^{2})/2!] + [(x^{4})/4!]NOTE: y in this problem is a truncated version of the series representation of sin(x).

Find the derivative of y = 1 − [(x

^{2})/2!] + [(x^{4})/4!] − [(x^{6})/6!]- dy = [d/dx] (1 − [(x
^{2})/2!] + [(x^{4})/4!] − [(x^{6})/6!]) - = 0 − [2x/2!] + [(4x
^{3})/4!] − [(6x^{5})/6!]

−x + [(x

NOTE: y in this problem is a truncated version of the series representation of cos(x).

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

Find the derivative of y = 1 + x + [(x

^{2})/2!] + [(x^{3})/3!] + [(x^{4})/4!]- dy = [d/dx] (1 + x + [(x
^{2})/2!] + [(x^{3})/3!] + [(x^{4})/4!])

1 + x + [(x

NOTE: y in this problem is a truncated version of the series representation of e

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

^{x}.*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.

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 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

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