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Possible mistake during differentiation? Please check

  1. Nov 20, 2011 #1
    For a function f(x), I have to determine intervals of increase/decrease, find local max(s)/min(s), and find intervals of concavity. The first thing I'm doing in this is to write out f'(x) and f''(x).

    f(x) = [itex]ln(x)/\sqrt{x}[/itex]

    For f'(x), I used the quotient rule and received f'(x) = (([itex]\frac{1}{x}[/itex][itex]\sqrt{x}[/itex])-([itex]\frac{-\sqrt{x}}{2}[/itex]ln(x))) / 2

    However, I plugged f(x) into wolfram alpha and it gave me: [itex]\frac{2-ln(x)}{2x^{3/2}}[/itex]

    I don't understand the difference? I thought I had done this correctly but apparently not? Wolfram alpha used the product rule. Is there some kind of algebraic gymnastics I'm forgetting about? I really want to understand where my error was made, not just which is the correct answer. Thanks!
  2. jcsd
  3. Nov 20, 2011 #2
    In using the quotient rule you are not differentiating [itex]\sqrt{x}[/itex] corretly and the quotient rule requires you to divide by the square of the denominator.
  4. Nov 20, 2011 #3


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    I see a couple of issues here:

    1) How did you get [itex]\frac{-\sqrt{x}}{2}[/itex] in the numerator? What is the derivative of [itex]\sqrt{x}[/itex] ?

    2) How did you get 2 in the denominator? [itex](\sqrt{x})^{2} = [/itex] ?
  5. Nov 20, 2011 #4
    Using quotient rule, you should get [itex]\frac{\frac{\sqrt{x}}{x} - \frac{\ln{x}}{2 \sqrt{x}}}{x}[/itex] which simplifies to what you got from WA. It looks like you messed up on the derivative of [itex]\sqrt{x}[/itex] and on the bottom of the quotient rule. http://en.wikipedia.org/wiki/Quotient_rule
  6. Nov 20, 2011 #5
    The following is my logic for the answer I received:
    1. [itex]\sqrt{x}[/itex] = x[itex]^{1/2}[/itex]
    2. Using the power rule I bring the 1/2 out as a coefficient, and subtract one from the numerator : [itex]\frac{1}{2}[/itex][itex]x^{-1/2}[/itex]
    3. I simplified to : [itex]\frac{-\sqrt{x}}{2}[/itex]

    My bad! I did a poor job transcribing it from my notebook to the syntax used on this site. It was (obviously) my first post, but far from my last! I meant to put "x" as the denominator, that was a mental slip.
  7. Nov 20, 2011 #6
    Welcome to the PhysicsForums! One may sometimes swallow some water but that is the way to learns to swim!
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