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Find the derivative of arcsin(sin x)

by RoRi

Find the derivative of the function

    \[ f(x) = \arcsin (\sin x). \]

Using the chain rule and the formula for the derivative of arcsine we have

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

This is valid for all x such that \cos x \neq 0.

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