$$\text { Find the derivative of } y \text { with respect to } x , t , \text { or } \theta \text {, as appropriate. }$$
-$$y = \ln \left( 7 \theta e ^ { - \theta } \right)$$
A) $e ^ { \theta \left( \frac { 1 } { \theta } + 1 \right) }$
B) $\ln \left( 7 e ^ { - \theta } ( 1 - \theta ) \right)$
C) $\frac { 1 } { 7 \theta e ^ { \theta } }$
D) $\frac { 1 } { \theta } - 1$

Question

Quizplus · Accepted Answer

Using the chain rule, we have:

$$ \frac{ dy } { d \theta } = \frac { 1 } { 7 \theta e ^ { -\theta } } \cdot \frac { d } { d \theta } \left( 7 \theta e ^ { -\theta } \right)$$
$$ = \frac { 1 } { 7 \theta e ^ { -\theta } } \cdot \left( 7 e ^ { -\theta } - 7 \theta e ^ { -\theta } \cdot ( -1 ) \right) $$
$$ = \frac { 1 } { 7 \theta e ^ { -\theta } } \cdot \left( 7 e ^ { -\theta } + 7 \theta e ^ { -\theta } \right) $$
$$ = \frac { 1 } { \theta } $$

Thus, the answer is D, $\frac{1}{\theta} - 1$.

Find the derivative of y with respect to x,t, or θ, as appropriate. \text { Find the derivative of } y \text { with respect to } x , t , \text { or } \theta \text {, as appropriate. } Find the derivative of y with respect to x,t, or θ, as appropriate.

$\text { Find the derivative of } y \text { with respect to } x , t , \text { or } \theta \text {, as appropriate. }$