Find the derivative of the function
![\[ f(x) = \arctan(\tan^2 x). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1443c72fc60f055c1c9d1ffb1ee0d4bc_l3.png "Rendered by QuickLaTeX.com")
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*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-b2fe6f0850adde845eb306c07902f974_l3.png "Rendered by QuickLaTeX.com")
This formula is valid everywhere 
is defined (i.e., everywhere 
).
Find the derivative of the function
Using the formula for the derivative of the tangent and arctangent we compute,
This formula is valid everywhere is defined (i.e., everywhere ).