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Need help with quotient rule

  1. Jan 11, 2012 #1
    1. The problem statement, all variables and given/known data

    I want to prove that if [tex] y = \frac{u}{v} [/tex]

    then [tex] \frac{dy}{dx} = \frac{ v \frac{du}{dx} - u \frac{dv}{dx} }{v²} [/tex]

    u and v are functions of x.

    2. The attempt at a solution

    [tex] y = uv^{-1} [/tex]

    [tex] y + dy = ( u + du ) ( v + dv )^{-1} [/tex]

    then I suppose I could use Newton's Binomial to develop

    [tex] ( v + dv )^{-1} [/tex]

    but I don't know how to use the formula

    [tex] (a+b)^{n} = \sum_{k=0}^{n} \dbinom{n}{k} a^{n-k} b^k [/tex]

    with a negative exponent. I'm familiar with binomial coefficients but that negative exponent is leaving me without a clue.

    Any help would be very much appreciated, thank you!
     
    Last edited: Jan 11, 2012
  2. jcsd
  3. Jan 11, 2012 #2

    Char. Limit

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    Gold Member

    That seems like a rather awkward way to do it. Are you allowed to use the product rule in your proof?
     
  4. Jan 11, 2012 #3
    I am learning by myself for now, so I'm pretty much allowed to use anything.
    How would you do it using the product rule?
     
  5. Jan 11, 2012 #4

    Char. Limit

    User Avatar
    Gold Member

    Well, if you can use the product rule, it becomes a lot easier. Since the product rule is:

    (uv)' = u v' + u' v

    Just say (u * 1/v)' = u (1/v)' + u' (1/v)

    and solve from there.
     
  6. Jan 11, 2012 #5
    Oooh this is indeed much simpler. Thank you!
     
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