1 It is given that \(\mathrm { f } ( x ) = \tan ^ { - 1 } 2 x\) and \(\mathrm { g } ( x ) = p \tan ^ { - 1 } x\), where \(p\) is a constant. Find the value of \(p\) for which \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 2 } \right) = \mathrm { g } ^ { \prime } \left( \frac { 1 } { 2 } \right)\).