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Thread: Quotient rule for derivatives

  1. #1
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    Question Quotient rule for derivatives

    I am trying to use the quotient rule on this:

    $\displaystyle % MathType!MTEF!2!1!+-
    % faaagaart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedm vETj
    % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpe pC0x
    % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq
    % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa
    % aiaabeqaamaabaabaaGcbaWaaSaaaeaadaGcaaqaaiaadIhaaS qaba
    % aakeaadaGcaaqaaiaadMhaaSqabaaaaaaa!305D!
    \[
    \frac{{\sqrt x }}{{\sqrt y }}
    \]
    $

    I was taught the quotient rule as:

    $\displaystyle % MathType!MTEF!2!1!+-
    % faaagaart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedm vETj
    % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpe pC0x
    % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq
    % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa
    % aiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGMbGaai4jaiaadE gacq
    % GHsislcaWGMbGaam4zaiaacEcaaeaacaWGNbWaaWbaaSqabeaa caaI
    % Yaaaaaaaaaa!35E8!
    \[
    \frac{{f'g - fg'}}{{g^2 }}
    \]
    $ which has been working fine until I hit this question while studying at home.

    I don't know whether to take the derivative of the denominator as the derivative of $\displaystyle % MathType!MTEF!2!1!+-
    % faaagaart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedm vETj
    % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpe pC0x
    % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq
    % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa
    % aiaabeqaamaabaabaaGcbaGaamyEamaaCaaaleqabaGaaGymai aac+
    % cacaaIYaaaaaaa!3167!
    \[
    y^{1/2}
    \]
    $ or the derivative of $\displaystyle % MathType!MTEF!2!1!+-
    % faaagaart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedm vETj
    % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpe pC0x
    % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq
    % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa
    % aiaabeqaamaabaabaaGcbaGaamyEamaaCaaaleqabaGaeyOeI0 IaaG
    % ymaiaac+cacaaIYaaaaaaa!3254!
    \[
    y^{ - 1/2}
    \]
    $ because it is in the denominator? I'm confused because I can rewrite the function as $\displaystyle % MathType!MTEF!2!1!+-
    % faaagaart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedm vETj
    % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpe pC0x
    % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq
    % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa
    % aiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGymai aac+
    % cacaaIYaaaaOGaamyEamaaCaaaleqabaGaeyOeI0IaaGymaiaa c+ca
    % caaIYaaaaaaa!35B2!
    \[
    x^{1/2} y^{ - 1/2}
    \]
    $. What is my g for this formula? Thanks in advance, gurus!!!
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  2. #2
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    Treat it as $\displaystyle \displaystyle y^{\frac{1}{2}}$.
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