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Compute the derivative of the given logarithmic function

by RoRi

Compute the derivative of the function

    \[ f(x) = \frac{1}{2 \sqrt{ab}} \log \left( \frac{\sqrt{a} + x \sqrt{b}}{\sqrt{a} - x \sqrt{b}} \right). \]

Using the rule for the derivative of the logarithm and the chain rule we compute directly,

    \begin{align*} f'(x) &= \frac{1}{2\sqrt{ab}} \left( \frac{\sqrt{a} - x \sqrt{b}}{\sqrt{a} + x \sqrt{b}} \right) \left( \frac{\sqrt{b}(\sqrt{a} - x \sqrt{b}) + \sqrt{b}(\sqrt{a} + x \sqrt{b})}{\left( \sqrt{a} - x\sqrt{b}\right)^2} \right) \\ &= \left( \frac{1}{2 \sqrt{ab}} \right) \left( \frac{\sqrt{a} - x \sqrt{b}}{\sqrt{a} + x \sqrt{b}} \right) \left( \frac{ \sqrt{b}(2 \sqrt{a})}{\left( \sqrt{a} - x\sqrt{b}\right)^2} \right) \\ &= \left( \frac{1}{2 \sqrt{ab}} \right) \left( \frac{2 \sqrt{ab}}{(\sqrt{a} + x \sqrt{b})(\sqrt{a} - x \sqrt{b})} \right) \\ &= \frac{1}{a^2 - bx^2}. \end{align*}

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