$$\text { Find the derivative of } y \text { with respect to } x , t , \text { or } \theta \text {, as appropriate. }$$
-$$y = \ln \left( \frac { e ^ { \theta } } { 5 + e ^ { \theta } } \right)$$
A) $\frac { 5 + e ^ { \theta } } { e ^ { \theta } }$
B) $\frac { 5 } { 5 + e ^ { \theta } }$
C) $\ln \left( \frac { 5 } { 5 + e ^ { \theta } } \right)$
D) $\frac { 5 + 2 e ^ { \theta } } { 5 + e ^ { \theta } }$

Question

Quizplus · Accepted Answer

To find the derivative of $y$ with respect to $\theta$, we apply the chain rule and the derivative of the natural logarithm function. The derivative of $\ln(u)$ is $\frac{1}{u}\frac{du}{d\theta}$, where $u = \frac{e^\theta}{5 + e^\theta}$. Differentiating $u$ with respect to $\theta$ gives $\frac{5 + e^\theta - e^\theta}{(5 + e^\theta)^2} = \frac{5}{(5 + e^\theta)^2}$. When we apply the chain rule, the $(5 + e^\theta)^2$ in the denominator cancels with one of the $(5 + e^\theta)$ in the numerator from the derivative of $\ln(u)$, leaving $\frac{5}{5 + e^\theta}$.

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.

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