Use the Properties of the Derivative to Find $D _ { t } ( \mathbf { r } ( t ) \times \mathbf { u } ( t ) ) \text { given the following }$

Multiple Choice

Use the properties of the derivative to find
$D _ { t } ( \mathbf { r } ( t ) \times \mathbf { u } ( t ) ) \text { given the following }$ vector-valued functions. $\begin{array} { l } \mathbf { r } ( t ) = 4 \cos 4 t \mathbf { j } + 4 \sin 4 t \mathbf { k } \\\mathbf { u } ( t ) = \frac { 5 } { t } \mathbf { i } + 4 \cos 4 t \mathbf { j } + 4 \sin 4 t \mathbf { k }\end{array}$

A) $\left( \frac { 80 \cos ( 4 t ) } { t } - \frac { 20 \sin ( 4 t ) } { t ^ { 2 } } \right) \mathbf { j } + \left( \frac { 20 \cos ( 4 t ) } { t ^ { 2 } } - \frac { 80 \sin ( 4 t ) } { t } \right) \mathbf { k }$
B) $\left( \frac { 80 \cos ( 4 t ) } { t } + \frac { 20 \sin ( 4 t ) } { t ^ { 2 } } \right) \mathbf { j } + \left( \frac { 20 \cos ( 4 t ) } { t ^ { 2 } } + \frac { 80 \sin ( 4 t ) } { t } \right) \mathbf { k }$
C) $\left( \frac { 80 \cos ( 4 t ) } { t ^ { 2 } } - \frac { 20 \sin ( 4 t ) } { t } \right) \mathbf { j } + \left( \frac { 20 \cos ( 4 t ) } { t ^ { 2 } } + \frac { 80 \sin ( 4 t ) } { t } \right) \mathbf { k }$
D) $\left( \frac { 80 \cos ( 4 t ) } { t } - \frac { 20 \sin ( 4 t ) } { t ^ { 2 } } \right) \mathbf { i } + \left( \frac { 20 \cos ( 4 t ) } { t ^ { 2 } } - \frac { 80 \sin ( 4 t ) } { t } \right) \mathbf { k }$
E) $\left( \frac { 80 \cos ( 4 t ) } { t } - \frac { 20 \sin ( 4 t ) } { t ^ { 2 } } \right) \mathbf { j } + \left( \frac { 20 \cos ( 4 t ) } { t ^ { 2 } } + \frac { 80 \sin ( 4 t ) } { t } \right) \mathbf { k }$