Answer
$\textbf{r}''(t) = \langle 16e^{4t},32e^{-4t}, 2e^{-t}\rangle$
$\textbf{r}'''(t) = \langle 64e^{4t},-128e^{-4t}, -2e^{-t}\rangle$
Work Step by Step
To find the derivative, simply find the derivative of each component.
$\textbf{r}(t) = \langle e^{4t},2e^{-4t}, 2e^{-t}\rangle$
$\textbf{r}'(t) = \langle 4e^{4t},-8e^{-4t}, -2e^{-t}\rangle$
$\textbf{r}''(t) = \langle 16e^{4t},32e^{-4t}, 2e^{-t}\rangle$
$\textbf{r}'''(t) = \langle 64e^{4t},-128e^{-4t}, -2e^{-t}\rangle$
