Find the Derivative of the Function $t ( x ) = \frac { \cot x } { 4 + 8 \sec x }$

question 67

Multiple Choice

Find the derivative of the function. $t ( x ) = \frac { \cot x } { 4 + 8 \sec x }$

• A) \(t ^ { \prime } ( x ) = - \frac { ( 4 - 8 \sec x ) \csc ^ { 2 } x + 8 \sec x } { ( 4 - 8 \sec x ) ^ { 2 } }\)
• B) \(t ^ { \prime } ( x ) = - \frac { ( 4 + 8 \sec x ) \csc ^ { 2 } x + 8 \sec x } { ( 4 + 8 \sec x ) ^ { 2 } }\)
• C) \(t ^ { \prime } ( x ) = \frac { ( 4 + 8 \sec x ) \csc ^ { 2 } x + 8 \sec x } { ( 4 + 8 \sec x ) ^ { 2 } }\)
• D) \(t ^ { \prime } ( x ) = - \frac { ( 4 + 8 \sec x ) \csc ^ { 2 } x + 8 \sec x } { ( 4 - 8 \sec x ) ^ { 2 } }\)
• E) \(t ^ { \prime } ( x ) = - \frac { ( 4 - 8 \sec x ) \csc ^ { 2 } x - 8 \sec x } { ( 4 - 8 \sec x ) ^ { 2 } }\)

---

Definitions:

Utilitarianism

A moral theory that advocates for actions that maximize happiness and well-being for the greatest number of people.

Liberalism

A belief system centered on the principles of freedom, the governed's agreement, and equal treatment under the law.

Libertarianism

A political philosophy that emphasizes freedom, autonomy, and voluntary association.

Diminishing Marginal Utility

An economic theory suggesting that the utility or satisfaction derived from consuming more of a good decreases with each subsequent unit.