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Find the derivative of log (arccos (x-1/2))

by RoRi

Find the derivative of the function

    \[ f(x) = \log \left( \arccos \frac{1}{\sqrt{x}} \right). \]

Using the chain rule and the formulas for the derivative of the logarithm and the arccosine we have,

    \begin{align*} f'(x) &= \left( \frac{1}{\arccos \left( \frac{1}{\sqrt{x}} \right)} \right) \left( \frac{-1}{\sqrt{1 - \left( \frac{1}{\sqrt{x}} \right)^2}} \right) \left(\frac{-1}{2x^{\frac{3}{2}}} \right)\\ &= \frac{\sqrt{x}}{2x^{\frac{3}{2}} \sqrt{x-1} \arccos \frac{1}{\sqrt{x}}} \\ &= \frac{1}{2x \sqrt{x-1} \arccos \frac{1}{\sqrt{x}}}. \end{align*}

The formula is valid for x > 1.

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