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Find the derivative of arctan (x+(1+x2)1/2)

by RoRi

Find the derivative of the function

    \[ f(x) = \arctan \left( x + \sqrt{1+x^2} \right). \]

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

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

This is valid for all x.

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