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) $\left( \frac { 80 \cos ( 4 t ) } { t } - \frac { 20 \sin ( 4 t ) } { t ^ { 2 } } \right) \mathbf { j } + \left( \frac { 20 \cos ( 4 t ) } { t ^ { 2 } } + \frac { 80 \sin ( 4 t ) } { t } \right) \mathbf { k }$

Question

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) $\left( \frac { 80 \cos ( 4 t ) } { t } - \frac { 20 \sin ( 4 t ) } { t ^ { 2 } } \right) \mathbf { j } + \left( \frac { 20 \cos ( 4 t ) } { t ^ { 2 } } + \frac { 80 \sin ( 4 t ) } { t } \right) \mathbf { k }$

Quizplus · Accepted Answer

The derivative of the cross product of two vector-valued functions, $\mathbf{r}(t) \times \mathbf{u}(t)$, can be found using the product rule for derivatives, which states $D_t(\mathbf{r}(t) \times \mathbf{u}(t)) = D_t(\mathbf{r}(t)) \times \mathbf{u}(t) + \mathbf{r}(t) \times D_t(\mathbf{u}(t))$. After calculating the derivatives of $\mathbf{r}(t)$ and $\mathbf{u}(t)$ and applying the cross product, the correct answer involves terms that correctly reflect the derivatives and cross product operations, leading to the components listed in option E.

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

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