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Math Help - Quotient rule for derivatives

  1. #1
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    Apr 2011
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    6

    Question Quotient rule for derivatives

    I am trying to use the quotient rule on this:

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

    I was taught the quotient rule as:

    % MathType!MTEF!2!1!+-<br />
% faaagaart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedm  vETj<br />
% 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpe  pC0x<br />
% bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq<br />
% pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa<br />
% aiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGMbGaai4jaiaadE  gacq<br />
% GHsislcaWGMbGaam4zaiaacEcaaeaacaWGNbWaaWbaaSqabeaa  caaI<br />
% Yaaaaaaaaaa!35E8!<br />
\[<br />
\frac{{f'g - fg'}}{{g^2 }}<br />
\]<br />
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 % MathType!MTEF!2!1!+-<br />
% faaagaart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedm  vETj<br />
% 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpe  pC0x<br />
% bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq<br />
% pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa<br />
% aiaabeqaamaabaabaaGcbaGaamyEamaaCaaaleqabaGaaGymai  aac+<br />
% cacaaIYaaaaaaa!3167!<br />
\[<br />
y^{1/2} <br />
\]<br />
or the derivative of % MathType!MTEF!2!1!+-<br />
% faaagaart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedm  vETj<br />
% 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpe  pC0x<br />
% bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq<br />
% pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa<br />
% aiaabeqaamaabaabaaGcbaGaamyEamaaCaaaleqabaGaeyOeI0  IaaG<br />
% ymaiaac+cacaaIYaaaaaaa!3254!<br />
\[<br />
y^{ - 1/2} <br />
\]<br />
because it is in the denominator? I'm confused because I can rewrite the function as % MathType!MTEF!2!1!+-<br />
% faaagaart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedm  vETj<br />
% 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpe  pC0x<br />
% bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq<br />
% pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa<br />
% aiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGymai  aac+<br />
% cacaaIYaaaaOGaamyEamaaCaaaleqabaGaeyOeI0IaaGymaiaa  c+ca<br />
% caaIYaaaaaaa!35B2!<br />
\[<br />
x^{1/2} y^{ - 1/2} <br />
\]<br />
. 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 y^{\frac{1}{2}}.
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