Answer
$$\frac{{dy}}{{dt}} = \frac{1}{{2{t^{1/2}}{{\left( {t + 1} \right)}^{3/2}}}}$$
Work Step by Step
$$\eqalign{
& y = \sqrt {\frac{t}{{t + 1}}} \cr
& y = {\left( {\frac{t}{{t + 1}}} \right)^{1/2}} \cr
& {\text{Take the natural logarithm of both sides:}} \cr
& {\text{Use the properties of logarithms}} \cr
& \ln y = \ln {\left( {\frac{t}{{t + 1}}} \right)^{1/2}} \cr
& {\text{power rule:}} \cr
& \ln y = \frac{1}{2}\ln \left( {\frac{t}{{t + 1}}} \right) \cr
& {\text{quotient rule:}} \cr
& \ln y = \frac{1}{2}\ln \left( t \right) - \frac{1}{2}\ln \left( {t + 1} \right) \cr
& {\text{Take derivatives of both sides with respect to }}t \cr
& \frac{1}{y}\frac{{dy}}{{dt}} = \frac{1}{2}\left( {\frac{1}{t}} \right) - \frac{1}{2}\left( {\frac{1}{{t + 1}}} \right) \cr
& \frac{1}{y}\frac{{dy}}{{dt}} = \frac{1}{2}\left( {\frac{1}{t} - \frac{1}{{t + 1}}} \right) \cr
& {\text{solve for }}\frac{{dy}}{{dt}} \cr
& \frac{{dy}}{{dt}} = \frac{y}{2}\left( {\frac{1}{t} - \frac{1}{{t + 1}}} \right) \cr
& \frac{{dy}}{{dt}} = \frac{y}{2}\left( {\frac{{t + 1 - t}}{{t\left( {t + 1} \right)}}} \right) \cr
& \frac{{dy}}{{dt}} = \frac{y}{2}\left( {\frac{1}{{t\left( {t + 1} \right)}}} \right) \cr
& {\text{substitute }}{\left( {\frac{t}{{t + 1}}} \right)^{1/2}}{\text{ for }}y{\text{ }} \cr
& \frac{{dy}}{{dt}} = \frac{1}{2}{\left( {\frac{t}{{t + 1}}} \right)^{1/2}}{\text{ }}\left( {\frac{1}{{t\left( {t + 1} \right)}}} \right) \cr
& \frac{{dy}}{{dt}} = \frac{1}{{2{t^{1/2}}{{\left( {t + 1} \right)}^{3/2}}}} \cr} $$
