Answer
\[\mathbf{r}'\left( t \right)=4{{e}^{t}}\mathbf{i}+\frac{1}{t}\mathbf{k}\]
Work Step by Step
\[\begin{align}
& \mathbf{r}\left( t \right)=4{{e}^{t}}\mathbf{i}+5\mathbf{j}+\ln t\mathbf{k} \\
& \text{Differentiate} \\
& \mathbf{r}'\left( t \right)=\frac{d}{dt}\left[ 4{{e}^{t}}\mathbf{i}+5\mathbf{j}+\ln t\mathbf{k} \right] \\
& \mathbf{r}'\left( t \right)=\frac{d}{dt}\left[ 4{{e}^{t}}\mathbf{i} \right]+\frac{d}{dt}\left[ 5\mathbf{j} \right]+\frac{d}{dt}\left[ \ln t\mathbf{k} \right] \\
& \mathbf{r}'\left( t \right)=4{{e}^{t}}\mathbf{i}+0\mathbf{j}+\frac{1}{t}\mathbf{k} \\
& \mathbf{r}'\left( t \right)=4{{e}^{t}}\mathbf{i}+\frac{1}{t}\mathbf{k} \\
\end{align}\]
