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Different results with quotient rule

  1. May 26, 2013 #1
    I have been trying to figure out the derivative of (X²-1)/X. When I use the quotient rule, the result I get is 1-1/X². However, when I simplify the expression first, then take the derivative, I get 1+1/X²

    Why are the results different?
     
  2. jcsd
  3. May 26, 2013 #2
  4. May 26, 2013 #3
    Can you show how you evaluate it? I tried both ways, but get different results.
     
  5. May 26, 2013 #4
    Alright, for a function
    [tex]f(x) = \frac{g(x)}{h(x)}[/tex]
    the quotient rule says
    [tex]f'(x) = \frac{g'(x)h(x)-g(x)h'(x)}{h^2(x)}[/tex]

    In your case, [itex]g(x)=x^2-1[/itex] and [itex]h(x)=x[/itex]. Thus [itex]g'(x)=2x[/itex] and [itex]h'(x)=1[/itex]. Then you get
    [tex]\frac{\mathrm{d}}{\mathrm{dx}}\frac{x^2-1}{x}=\frac{2x\cdot x - \left( x^2 -1 \right) \cdot 1}{x^2}=\frac{2x^2-x^2+1}{x^2}=1+\frac{1}{x^2}[/tex]
    Are you sure you included the [itex]g'(x)[/itex] term?

    Simplifying first,
    [tex]\frac{\mathrm{d}}{\mathrm{dx}}\frac{x^2-1}{x} = \frac{\mathrm{d}}{\mathrm{dx}} \left( x - \frac{1}{x} \right) = 1- \frac{-1}{x^2}=1+\frac{1}{x^2}[/tex]
     
  6. May 26, 2013 #5
    Where did +1 come from when you were multiplying x²-1 with 1?
     
  7. May 26, 2013 #6
    It was also multiplied by -1, from the quotient rule formula.
     
  8. May 26, 2013 #7
    I'm sorry for asking so many questions, and this may sound stupid, but, what -1?

    Edit: It appears as though I have made a mistake. The -1 is the minus part that is in front of the x²-1, so the negative, or minus, distributes itself.

    Thank you for the help, I appreciate it.
     
  9. May 26, 2013 #8
    You see the minus sign in front of [itex]\left( x^2 -1 \right) \cdot 1[/itex], right? That is just short notation for

    [tex]- \left( x^2 -1 \right) \cdot 1=\left( -1 \right) \cdot \left( x^2 -1 \right)= (-1) \cdot x^2 + (-1) \cdot (-1) = -x^2+1[/tex]

    Agreed? Or are you pulling my leg here?

    EDIT: Ah, good. I was almost giving up for a while there.
     
  10. May 26, 2013 #9

    SteamKing

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    Look at the formula for the quotient rule.

    If h(x) = x, what is h'(x)?
     
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