Understanding the Derivative of Arc Cosecant

The video tutorial provides a detailed proof of the derivative of f(x) = arc cosecant x. It begins by defining the variables and setting up the equation, followed by modeling the problem using a reference triangle. The tutorial then employs implicit differentiation to derive the equation, simplifying it to reach the final proof. The derivative is shown to be negative, indicating a decreasing function. The tutorial concludes with a graphical verification of the derivative's properties.

What is the derivative of arc cosecant x with respect to x?

• Negative 1 divided by the absolute value of x times the square root of x squared minus 1

• 1 divided by the square root of x squared minus 1

• Negative x divided by the square root of x squared minus 1

• x divided by the square root of x squared minus 1

In the context of arc cosecant, what does the variable y represent?

• The opposite side

• The cosecant function value

• The hypotenuse

• The angle

Why can't y equal zero in the context of arc cosecant?

• Because cosecant is undefined when sine is zero

• Because the angle must be positive

• Because the hypotenuse cannot be zero

• Because x cannot be negative

What is the relationship between cosecant and sine?

• They are unrelated

• They are equal

• Sine is the reciprocal of cosecant

• Cosecant is the reciprocal of sine

What is the range of the angle y in the context of arc cosecant?

• Negative pi to pi

• Negative pi over two to positive pi over two, excluding zero

• 0 to pi

• 0 to 2 pi

What is the derivative of cosecant y with respect to x?

• Negative sine y times tangent y

• Negative cosecant y times cotangent y times dy/dx

• Cosecant y times cotangent y

• Sine y times tangent y

What substitution is made for sine y during the simplification?

• The absolute value of x

• The square root of x squared minus 1

• x divided by 1

• 1 divided by x

Why is it important to ensure the remaining factor of x in the denominator is positive?

• To maintain the integrity of the derivative

• To ensure the hypotenuse is positive

• To avoid division by zero

• To ensure the function is increasing

What does the graph of y = arc cosecant x demonstrate about the function?

• The slope of the tangent line is always positive

• The function is always increasing

• The function is constant

• The slope of the tangent line is always negative

What is the significance of the derivative being negative?

• The function is increasing

• The function is decreasing

• The function is constant

• The function is undefined