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More Help with Derivatives

  1. Jul 2, 2012 #1
    I'll just make one thread for all the help I'll need with derivatives so I don't clutter up this forum.

    1. The problem statement, all variables and given/known data
    Find the derivative of y = sqrt(x)(x - 1).


    2. Relevant equations
    Wolfram Alpha gets this:
    http://www.wolframalpha.com/input/?i=derivative+y+=+sqrt(x)(x+-+1)

    I got sqrt(x) + [(x - 1) / (2sqrt(x))]. Which is basically everything up until the point where Wolfram returns the answer.

    I don't understand where the 3 in the numerator comes from, or where the sqrt(x) that's being added goes.

    3. The attempt at a solution
    All the work you see Wolfram doing, up until the point Wolfram returns the answer.
     
  2. jcsd
  3. Jul 2, 2012 #2

    HallsofIvy

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    Science Advisor

    It appears that you have used the product rule: the derivative of [itex]x^{1/2}(x-1)[/itex] is [itex](1/2)x^{-1/2}(x- 1)+ x^{1/2}(1)= \sqrt{x}+ (x- 1)/2\sqrt{x}[/itex].

    However, you can also write [itex]x^{1/2}(x- 1)= x^{3/2}- x^{1/2}[/itex]. Then the derivative is [itex](3/2)x^{1/2}- (1/2)x^{-1/2}[/itex]. That is what Wolfram is doing.

    Of course, those are the same. In the your answer, [itex]\sqrt{x}+ (x-1)/2\sqrt{x}[/itex], [itex]x/\sqrt{x}= \sqrt{x}[/itex] so that can be written [itex]\sqrt{x}+ (1/2)\sqrt{x}- 1/2\sqrt{x}= (3/2)\sqrt{x}- (1/2)x^{-1/2}[/itex], the same as Wolfram's answer.
     
  4. Jul 2, 2012 #3

    eumyang

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    Homework Helper

    Wolfram added the fractions together by finding the LCD.
    [tex]\sqrt{x} + \frac{x - 1}{2\sqrt{x}}[/tex]
    The 1st "fraction" has a denominator of 1, so the LCD is 2 sqrt (x). Multiply top and bottom of the 1st "fraction" by this LCD:
    [tex]\frac{\sqrt{x} \cdot 2\sqrt{x}}{1\cdot 2\sqrt{x}} + \frac{x - 1}{2\sqrt{x}}[/tex]
    I'll let you figure out the rest.
     
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