The derivative of ${\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right)$ with respect to ${\tan ^{ - 1}}\left( {{{2x\sqrt {1 - {x^2}} } \over {1 - 2{x^2}}}} \right)$ at x = ${1 \over 2}$ is :

Question

The derivative of
${\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right)$ with
respect to ${\tan ^{ - 1}}\left( {{{2x\sqrt {1 - {x^2}} } \over {1 - 2{x^2}}}} \right)$ at x = ${1 \over 2}$ is :

${{2\sqrt 3 } \over 3}$

${{2\sqrt 3 } \over 5}$

${{\sqrt 3 } \over {10}}$

${{\sqrt 3 } \over {12}}$

Solution

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