Answer
$${\bf{r}}{\text{'}}\left( t \right) = \frac{2}{{\sqrt t }}{\bf{i}} + \frac{5}{2}t\sqrt t {\bf{j}} + \frac{2}{t}{\bf{k}}$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = 4\sqrt t {\bf{i}} + {t^2}\sqrt t {\bf{j}} + \ln {t^2}{\bf{k}} \cr
& {\bf{r}}{\text{'}}\left( t \right) = \frac{d}{{dt}}\left[ {4\sqrt t } \right]{\bf{i}} + \frac{d}{{dt}}\left[ {{t^2}\sqrt t } \right]{\bf{j}} + \frac{d}{{dt}}\left[ {\ln {t^2}} \right]{\bf{k}} \cr
& {\bf{r}}{\text{'}}\left( t \right) = 4\frac{d}{{dt}}\left[ {{t^{1/2}}} \right]{\bf{i}} + \frac{d}{{dt}}\left[ {{t^{5/2}}} \right]{\bf{j}} + 2\frac{d}{{dt}}\left[ {\ln t} \right]{\bf{k}} \cr
& {\text{Differentiating}} \cr
& {\bf{r}}{\text{'}}\left( t \right) = 4\left( {\frac{1}{{2\sqrt t }}} \right){\bf{i}} + \left( {\frac{5}{2}{t^{3/2}}} \right){\bf{j}} + 2\left( {\frac{1}{t}} \right){\bf{k}} \cr
& {\bf{r}}{\text{'}}\left( t \right) = \frac{2}{{\sqrt t }}{\bf{i}} + \frac{5}{2}t\sqrt t {\bf{j}} + \frac{2}{t}{\bf{k}} \cr} $$
