Answer
$$f'\left( t \right) = \frac{1}{{\left( {1 - t} \right)\sqrt t }}$$
Work Step by Step
$$\eqalign{
& f\left( t \right) = 2{\tanh ^{ - 1}}\sqrt t \cr
& {\text{find the derivative}} \cr
& f'\left( t \right) = \frac{d}{{dt}}\left( {2{{\tanh }^{ - 1}}\sqrt t } \right) \cr
& f'\left( t \right) = 2\frac{d}{{dt}}\left( {{{\tanh }^{ - 1}}\sqrt t } \right) \cr
& {\text{use derivatives of the inverse hyperbolic functions}} \cr
& f'\left( t \right) = 2\left( {\frac{1}{{1 - {{\left( {\sqrt t } \right)}^2}}}} \right)\frac{d}{{dt}}\left( {\sqrt t } \right) \cr
& f'\left( t \right) = 2\left( {\frac{1}{{1 - {{\left( {\sqrt t } \right)}^2}}}} \right)\left( {\frac{1}{{2\sqrt t }}} \right) \cr
& {\text{simplify}} \cr
& f'\left( t \right) = \left( {\frac{1}{{1 - t}}} \right)\left( {\frac{1}{{\sqrt t }}} \right) \cr
& f'\left( t \right) = \frac{1}{{\left( {1 - t} \right)\sqrt t }} \cr} $$
