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Solving for a derivative. Website discrepancy to clarify?

Posted 9 years ago

I was looking for a derivative for the following function:

((x^3)-(3x^2)+(2sqrt(x))-5)/(sqrt(x))

However, the given solution on the first page does not match with the step by step solution provided. I want to understand why there is a difference in the final answer for each...is it that the step by step solution is actually incomplete, or is it that the provided solution on the main page is wrong?

Here is the link in regards to this:

http://www.wolframalpha.com/input/?i=derivative+%28%28x%5E3%29-%283x%5E2%29%2B%282sqrt%28x%29%29-5%29%2F%28sqrt%28x%29%29

POSTED BY: Komul Bedi
2 Replies

The plain text version simplifies the solution one more time to give Alpha's original answer:

Possible derivation:
d/dx((x^3-3 x^2+2 sqrt(x)-5)/sqrt(x))
Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = 1/sqrt(x) and v = x^3-3 x^2+2 sqrt(x)-5:
  =  (-5+2 sqrt(x)-3 x^2+x^3) (d/dx(1/sqrt(x)))+(d/dx(-5+2 sqrt(x)-3 x^2+x^3))/sqrt(x)
Use the power rule, d/dx(x^n) = n x^(n-1), where n = -1/2: d/dx(1/sqrt(x)) = d/dx(x^(-1/2)) = -1/2 x^(-3/2):
  =  (d/dx(-5+2 sqrt(x)-3 x^2+x^3))/sqrt(x)+(-5+2 sqrt(x)-3 x^2+x^3) (-1)/(2 x^(3/2))
Differentiate the sum term by term and factor out constants:
  =  -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+d/dx(-5)+2 d/dx(sqrt(x))-3 d/dx(x^2)+d/dx(x^3)/sqrt(x)
The derivative of -5 is zero:
  =  -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(2 (d/dx(sqrt(x)))-3 (d/dx(x^2))+d/dx(x^3)+0)/sqrt(x)
Simplify the expression:
  =  -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(2 (d/dx(sqrt(x)))-3 (d/dx(x^2))+d/dx(x^3))/sqrt(x)
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 1/2: d/dx(sqrt(x)) = d/dx(x^(1/2)) = x^(-1/2)/2:
  =  -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(-3 (d/dx(x^2))+d/dx(x^3)+2 1/(2 sqrt(x)))/sqrt(x)
Simplify the expression:
  =  -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(1/sqrt(x)-3 (d/dx(x^2))+d/dx(x^3))/sqrt(x)
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 2: d/dx(x^2) = 2 x:
  =  -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(1/sqrt(x)+d/dx(x^3)-3 2 x)/sqrt(x)
Simplify the expression:
  =  -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(1/sqrt(x)-6 x+d/dx(x^3))/sqrt(x)
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 3: d/dx(x^3) = 3 x^2:
  =  -(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))+(1/sqrt(x)-6 x+3 x^2)/sqrt(x)
Simplify the expression:
Answer: |  
 |   =  (1/sqrt(x)-6 x+3 x^2)/sqrt(x)-(-5+2 sqrt(x)-3 x^2+x^3)/(2 x^(3/2))
POSTED BY: Marvin Ray Burns
Posted 9 years ago

Sorry I don't understand what you mean. What you are showing there is the answer provided under the "step by step solution" section. However, this does not match with the solution provided as the derivative on the main page.

Again, my question is, which is the ACTUAL right answer and are there any missing/additional steps required to get to it?

POSTED BY: Komul Bedi
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