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Find the derivative of arcsin ((1-x2)/(1+x2))

by RoRi

Find the derivative of the function

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

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

    \begin{align*} f'(x) &= \left( \frac{-4x}{(1+x^2)^2} \right) \left( \frac{1}{\sqrt{1- \left( \frac{1-x^2}{1+x^2} \right)^2}} \right) \\[9pt] &= \left( \frac{-4x}{(1+x^2)^2} \right) \left( \frac{1+x^2}{\sqrt{(1+x^2)^2 - (1-x^2)^2}} \right) \\[9pt] &= \frac{-4x}{(1+x^2)\sqrt{4x^2}} \\[9pt] &= - \frac{2x}{|x| (1+x^2)}. \end{align*}

This is valid for x \neq 0.

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