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:

Continuum Theories

Theories that suggest changes or phenomena occur gradually over a continuum rather than being divided into distinct categories.

Smoking Cessation

The process of discontinuing tobacco smoking, which can significantly reduce the risk of developing smoking-related diseases.

Implementation Intentions

A self-regulatory strategy focusing on planning specific actions to achieve goals or change behaviors by forming if-then plans.

Health Behavior

Actions individuals take that affect their health, including lifestyle choices, diet, exercise, and adherence to medical advice.