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Find the derivative of arctan (tan2 x)

Find the derivative of the function

    \[ f(x) = \arctan(\tan^2 x). \]


Using the formula for the derivative of the tangent and arctangent we compute,

    \begin{align*}  f'(x) &= (2 \tan x \sec^2 x) \left( \frac{1}{1+\tan^4 x} \right) \\[9pt]  &= \frac{2 \frac{\sin x}{\cos^3 x} \cos^4 x}{\sin^4 x + \cos^4 x} \\[9pt]  &= \frac{2 \sin x \cos x}{\sin^4 x + \cos^4 x} \\[9pt]  &= \frac{\sin (2x)}{\sin^4 x + \cos^4 x}.  \end{align*}

This formula is valid everywhere \tan x is defined (i.e., everywhere \cos x \neq 0).

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