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Compute the derivative of (1/4) log((x2-1)/(x2+1))

Compute the derivative of

    \[ f(x) = \frac{1}{4} \log \frac{x^2-1}{x^2+1}. \]


First, using the properties of the logarithm we write,

    \[ f(x) = \frac{1}{4} (\log (x^2-1) - \log(x^2+1)). \]

Using the chain rule (and product rule) we have,

    \begin{align*}  f'(x) &= \frac{1}{4} \left( \frac{2x}{x^2-1} - \frac{2x}{x^2+1} \right) \\  &= \frac{1}{4} \left( \frac{2x(x^2+1) - 2x(x^2-1)}{(x^2-1)(x^2+1)} \right) \\  &= \frac{1}{4} \left( \frac{4x}{x^4-1} \right) \\  &= \frac{x}{x^4-1}. \end{align*}

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