SOLUTION: Differentiate: y = x^3 - 2x^2 / cubic root x^2 What I have done: y = x^3 - 2x^2 / x^2/3 dy/dx = x^2/3 (3x^2 - 4x) - (x^3 - 2x^2) (2/3x^-1/3) = 3x^8/3 - 4x^5/3 - 2/

Algebra ->  Expressions -> SOLUTION: Differentiate: y = x^3 - 2x^2 / cubic root x^2 What I have done: y = x^3 - 2x^2 / x^2/3 dy/dx = x^2/3 (3x^2 - 4x) - (x^3 - 2x^2) (2/3x^-1/3) = 3x^8/3 - 4x^5/3 - 2/      Log On


   

Question 696809: Differentiate:
y = x^3 - 2x^2 / cubic root x^2
What I have done:
y = x^3 - 2x^2 / x^2/3
dy/dx
= x^2/3 (3x^2 - 4x) - (x^3 - 2x^2) (2/3x^-1/3)
= 3x^8/3 - 4x^5/3 - 2/3x^8/3 + 4/3x^-2/3 / x^4/3
= 7/3x - 4x^5/3 + 4/3x^-2/3 / x^4/3
= 7/3x^-1/3 - 4x^1/3 + 4/3x^-2
Is my answer correct? Please let me know within 2 days as I have exam and I am unable to visit my teachers / classmates for help. Thanks!
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!

y%22%22=%22%22%28x%5E3+-+2x%5E2%29%2F%28root%283%2Cx%5E2%29%29

Change the cube root to a fraction exponent:

y%22%22=%22%22%28x%5E3+-+2x%5E2%29%2F%28x%5E%28%222%2F3%22%29%29

Change the denominator to a multiplication by a
negative exponential: 

matrix%282%2C1%2C%22%22%2Cy%29matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29matrix%282%2C1%2C%22%22%2C%28x%5E3-2x%5E2%29%28x%5E%28-2%2F3%29%29%29 

Distribute by adding exponents:

matrix%282%2C1%2C%22%22%2Cy%29matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29matrix%282%2C1%2C%22%22%2Cx%5E%283-2%2F3%29-%22%22%29matrix%282%2C1%2C%22%22%2C2x%5E%282-2%2F3%29%29

Get LCD's of fraction exponents

matrix%282%2C1%2C%22%22%2Cy%29matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29matrix%282%2C1%2C%22%22%2Cx%5E%289%2F3-2%2F3%29-%22%22%29matrix%282%2C1%2C%22%22%2C2x%5E%286%2F3-2%2F3%29%29 

Do the addition of the fraction exponents:

matrix%282%2C1%2C%22%22%2Cy%29matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29matrix%282%2C1%2C%22%22%2Cx%5E%287%2F3%29-%22%22%29matrix%282%2C1%2C%22%22%2C2x%5E%284%2F3%29%29

Now we can just use the basic power rule to
find the derivative:

matrix%282%2C1%2C%22%22%2Cdy%2Fdx%29matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29matrix%282%2C1%2C%22%22%2Cexpr%287%2F3%29x%5E%284%2F3%29-%22%22%29matrix%282%2C1%2C%22%22%2Cexpr%288%2F3%29x%5E%281%2F3%29%29

You can either leave it like that or factor out matrix%282%2C1%2C%22%22%2Cexpr%281%2F3%29x%5E%281%2F3%29%29 

matrix%282%2C1%2C%22%22%2Cdy%2Fdx%29matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29matrix%282%2C1%2C%22%22%2Cexpr%281%2F3%29x%5E%281%2F3%29%29matrix%282%2C1%2C%22%22%2C%22%22%2A%22%22%29matrix%282%2C1%2C%22%22%2C%287x%5E%283%2F3%29-8%29%29

Change the exponent 3%2F3 to 1 and the fractional exponential
to a cube root:

dy%2Fdx%22%22=%22%221%2F3root%283%2Cx%29%287x-8%29

I'm sure you can skip some of those steps.  I just broke it down 
all the way just in case.

Edwin