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Find the derivative of (ex – e-x)/(ex + e-x)

by RoRi

Find the derivative of the following function:

    \[ f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}. \]

We can compute this derivative directly using the quotient rule,

    \begin{align*} f'(x) &= \frac{(e^x + e^{-x})(e^x + e^{-x}) - (e^x - e^{-x})(e^x - e^{-x})}{(e^x+e^{-x})^2} \\[9pt] &= \frac{(e^{2x} + e^{-2x} + 2) - (e^{2x} + e^{-2x} - 2)}{(e^x+e^{-x})^2} \\[9pt] &= \frac{4}{(e^x+e^{-x})^2} \\[9pt] &= 4 (e^x + e^{-x})^{-2}. \end{align*}

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