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Simplifying a differentiation problem

  1. Sep 25, 2007 #1
    1. The problem statement, all variables and given/known data
    [3x^2 + 2(sqrt x)] / x

    My book says it simplified it to:

    3x + 2x ^-1/2


    I understand the 3x part since 3x^2/x = 3x
    But where did the second part come from? I tried to make sense with the power rule but it still doesn't seem clear to me. Anyone get it?
     
    Last edited: Sep 25, 2007
  2. jcsd
  3. Sep 25, 2007 #2

    dynamicsolo

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    Is this supposed to be [[3x^2 + 2(sqrt x)] / x ?

    Then it would make sense.
     
  4. Sep 25, 2007 #3
    Yes you're right. Can you explain it?
     
  5. Sep 25, 2007 #4
    There's a formula for simplying derivitives that look like big fractions like yours... i forget the name but the way it goes was: "((bottom) * derivitive[of the top] - (top)*derivitive[of the bottom])) over the bottom squared)"

    like this:

    (bottom * deriv[top]) - (top * deriv[bottom])
    ---------------------------------------------
    (bottom)^2

    that and square root is same as x^(1/2) power and just apply the chain rule i think it's called which makes it 1/(2*sqrt(x)).
    whats important is it's differant for this: (3x)^(1/2) which becomes (1/(2*sqrt(x)) * derivitive of (3x).

    on side note: if you dont have a ti89, get one. it does calculus and is extremely helpful in checking calculus homework and understanding why certain things differentiate the way they do.
     
  6. Sep 25, 2007 #5
    So this is at the end of the problem? Distribute the division: (a+b)/c = a/c + b/c
    Remember that sqrt(x) = x^1/2
     
  7. Sep 25, 2007 #6
    Yes I know the quotient rule and power rule, but I don't see how the first equation was simplified.
     
  8. Sep 25, 2007 #7
    There isn't a common factor in the original equation, though...
     
  9. Sep 25, 2007 #8

    hage567

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    I'm confused on what you think you are doing. Are you trying to differentiate the above expression, or just simplifying it? If you are just simplifying, you just divide the terms. No calculus involved in this step.
    So for the second term you have [tex]\frac{2\sqrt{x}}{x}[/tex], which is the same as [tex]\frac{2{x}^\frac{1}{2}}{x}[/tex]
    Do you know your laws of exponents?
     
  10. Sep 25, 2007 #9
    The book is saying that 2(sqrt 2) / x is the same as 2x^-1/2
    so while I did get the same answer as you, it apparently is not right according to the book.
     
  11. Sep 25, 2007 #10

    hage567

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    It is! [tex]\frac{2{x}^\frac{1}{2}}{x} = 2x^{\frac{1}{2}-1} = 2x^\frac{-1}{2}[/tex]

    When dividing, you subtract the exponents.

    I hope I'm understanding what you're asking.
     
  12. Sep 25, 2007 #11
    Oh I see it now...wow what a brain fart....anyways, thank you for explaining it to me clearly and in a nice manner.
     
  13. Sep 25, 2007 #12

    hage567

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    lol you're welcome.
     
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