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Find the derivative of arctan x + (1/3) arctan (x3)

by RoRi

Find the derivative of the function

    \[ f(x) = \arctan x + \frac{1}{3} \arctan (x^3). \]

Using the formula for the derivative of arctangent and the chain rule we compute,

    \begin{align*} f'(x) &= \frac{1}{1+x^2} + \frac{1}{3} (3x^2) \left( \frac{1}{1+(x^3)^2} \right) \\[9pt] &= \frac{1}{1+x^2} + \frac{x^2}{1+x^6} \\[9pt] &= \frac{1+x^6+x^2+x^4}{(1+x^2)(1+x^6)} \\[9pt] &= \frac{(1+x^2)(1+x^4)}{(1+x^2)(1+x^6)} \\[9pt] &= \frac{1+x^4}{1+x^6}. \end{align*}

This is valid for all x.

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