Find the derivative of y with respect to the independent variable.
-$$y = \log _ { 2 } \left( \frac { \sin \theta \cos \theta } { e ^ { \theta } 8 ^ { \theta } } \right)$$
A) $\frac { 1 } { \ln 2 } ( \cot \theta - \tan \theta - \ln 8 - 1 )$
B) $\mathrm { e } ^ { 2 } \left( \cos \theta - \sin \theta - \mathrm { e } ^ { \theta } 8 ^ { \theta } \right)$
C) $\frac { 1 } { \ln 2 } ( \sec \theta \csc \theta - \ln 8 - 1 )$
D) $\frac { 1 } { \ln 2 } \left( \frac { e ^ { \theta } \theta ^ { \theta } } { \sin \theta \cos \theta } \right)$

Question

Quizplus · Accepted Answer

To find the derivative of $y$ with respect to $\theta$, we use the chain rule and properties of logarithms. The derivative of $\log_2(x)$ is $\frac{1}{x\ln(2)}$, and we apply the product rule and chain rule to the argument of the logarithm. Simplifying the expression leads to the correct answer, which involves the derivatives of $\sin(\theta)\cos(\theta)$ and $e^\theta 8^\theta$, resulting in terms that include $\cot(\theta) - \tan(\theta)$ and constants derived from the derivative of the exponential and power functions.

Find the Derivative of Y with Respect to the Independent y=log⁡2(sin⁡θcos⁡θeθ8θ)y = \log _ { 2 } \left( \frac { \sin \theta \cos \theta } { e ^ { \theta } 8 ^ { \theta } } \right)y=log2​(eθ8θsinθcosθ​)

Find the Derivative of Y with Respect to the Independent $y = \log _ { 2 } \left( \frac { \sin \theta \cos \theta } { e ^ { \theta } 8 ^ { \theta } } \right)$