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Scholar
Karma Points: 281
Respect (98%):
Date Posted: 7/23/2008 1:47:47 PM  Status: Live
First derivatives
Course Textbook Chapter Problem
Calculus N/A N/A N/A
Question Details:
What is the first derivative of   y = x (x+2)3?
Also, what is y ''?
 
Thank you for your help.
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Answers:

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Mentor
Karma Points: 607
Date Posted: 7/23/2008 1:51:58 PM  Status: Live
Asker's Rating: Somewhat Helpful   
Response:
use product rule and chain rule:

first derivative: y' = 3x(x + 2)2 + (x + 2)3
second derivative: y'' = 6x(x + 2) + 3(x + 2)2 + 3(x + 2)2

You can simplify if you want by pulling out the common factor (x + 2)

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Oracle
Karma Points: 54,734
Date Posted: 7/23/2008 1:53:15 PM  Status: Live
Asker's Rating: Somewhat Helpful   
Response:
y=x(x+2)3
y'=[x][3(x+2)2(1)]+[(x+2)3][(1)]
y'=[x][3(x+2)2]+[(x+2)3]
y'=3x(x+2)2+(x+2)3
y'=(x+2)2[3x+(x+2)]
y'=(x+2)2[4x+2]
-----------------------
y''=[(x+2)2][(4)]+[(4x+2)][2(x+2)1(1)]
y''=4(x+2)2+2(x+2)(4x+2)
y''=2(x+2)[2(x+2)+(4x+2)]
y''=2(x+2)[+2x+4+4x+2]
y''=2(x+2)[6x+6]
-----------------

 

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Oracle
Karma Points: 15,133
Date Posted: 7/23/2008 1:54:11 PM  Status: Live
Asker's Rating: Helpful   
Response:
you may use product rule and chain rule to find y' and y", but this question, you may rewrite y by expanding,
y = x(x3 + 6x2 + 12x + 8) = x4 + 6x3 + 12x2 + 8x
y' = 4x3 + 18x2 + 24x + 8
y" = 12x2 + 36x + 24

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Sage
Karma Points: 4,678
Date Posted: 7/23/2008 5:03:35 PM  Status: Live
Asker's Rating: Lifesaver   
Response:
Hi,

For this question, we have to use the product rule which states:

dy = y' = uv' + vu'

Therefore, we define u and v as follows:

u = x           u' = 1

v = (x+2)3     v' = 3(x+2)2

Now, apply the Rule to get:

3x(x+2)2 + (x+2)3 = y'

Now, we know the first derivative, let's find the second derivative by using the product rule and chain rule to get:

u = 3x      u' = 3
                                                                 +    3(x+2)2                     = y''
v = (x+2)2  v' = 2(x+2) = 2x+4

Now, apply the product rule to get:

3x ( 2x + 4 ) + 3(x+2)2 + 3(x+2)2 = y''

Simplify the above to get:

6x2 + 12x + 6(x+2)2 = y''

I hope that helps you out!  Please let me know if you have any other questions!
The VC
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