Understanding Derivatives and Trigonometric Functions

The video tutorial explains how to find the derivative of a function, specifically focusing on simplifying trigonometric functions to avoid using the quotient rule. It demonstrates the process of rewriting functions in terms of sine and cosine, finding derivatives of trigonometric functions, and evaluating the derivative at a specific point, x = pi/6. The tutorial concludes with a graphical representation of the function and its tangent line, highlighting the slope and rate of change.

What is the main goal of the problem presented in the video?

• To find the integral of a function

• To find the derivative of a function

• To solve a differential equation

• To graph a trigonometric function

Which trigonometric identity is used to simplify the function?

• Secant and cosecant identities

• Sine and tangent identities

• Sine and cosine identities

• Tangent and cotangent identities

What rule is avoided by simplifying the function using trigonometric identities?

• Power rule

• Quotient rule

• Chain rule

• Product rule

What is the derivative of cosecant x?

• Cosecant x times cotangent x

• Negative cosecant x times cotangent x

• Cosecant squared x

• Negative cosecant squared x

What is the value of cosecant π/6?

• 1/2

• 1

• 2

• √3

What is the value of cotangent π/6?

• 2

• 1

• 1/√3

• √3

What type of triangle is used to find the trigonometric values at π/6?

• Isosceles triangle

• Equilateral triangle

• 30-60-90 triangle

• 45-45-90 triangle

What is the slope of the tangent line at x = π/6?

• Approximately 8

• Approximately -8

• Approximately 11.464

• Approximately -11.464

What does a negative slope of the tangent line indicate about the function at x = π/6?

• The function is increasing

• The function is decreasing

• The function is constant

• The function has a maximum

What is the instantaneous rate of change of the function at x = π/6?

• The function value at x = π/6

• The integral at x = π/6

• The derivative at x = π/6

• The limit at x = π/6