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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 ex.

*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.

### 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