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# Find the derivative of y with respect to the given independent variable:$y=\log _{3}\left(\left(\frac{x+1}{x-1}\right)^{\ln 3}\right)$

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We will use the properties of logarithms to simplify the function first before we differentiate:

\begin{align*} y &= \log_3 \left(\dfrac{x + 1}{x - 1} \right)^{\ln 3}\\ &\stackrel{(4b)}= \ln 3\log_3 \left(\dfrac{x + 1}{x - 1} \right)\\ &\stackrel{(2b)}= \ln 3[\log_3 (x + 1) - \log_3 (x - 1)] \end{align*}

Now by Formula 8 on the above expression (on both terms), we have:

\begin{align*} y' &= \ln 3\left[ \dfrac{1}{(x + 1)\ln 3} \cdot 1 -\dfrac{1}{(x - 1)\ln 3} \cdot 1 \right]\\ &= \dfrac{1}{x + 1} - \dfrac{1}{x - 1}\\ &=\dfrac{(x -1) - (x + 1)}{(x + 1)(x -1)}\\ &=-\dfrac{2}{x^2 - 1} \end{align*}

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