Understanding the Derivative of Arccotangent Function

The video tutorial provides a detailed proof of the derivative of the function f(x) = arccotangent x. It begins by setting up the problem and modeling the angle y using a reference triangle. The tutorial then uses implicit differentiation to find the derivative, showing that it equals -1/(1 + x^2). The proof is finalized by discussing the properties of the derivative, such as its negativity and the function's monotonic decrease. Finally, the tutorial verifies the proof by examining the graph of the function, confirming that the slope of the tangent line is always negative.

What is the derivative of arccotangent x with respect to x?

• 1 / (1 + x^2)

• -1 / (1 + x^2)

• 1 / (1 - x^2)

• -1 / (1 - x^2)

In which quadrants can the angle y be when y = arccotangent x?

• First and second quadrants

• Second and fourth quadrants

• First and third quadrants

• Third and fourth quadrants

What is the relationship between sin y and the sides of the reference triangle?

• sin y = x / hypotenuse

• sin y = 1 / hypotenuse

• sin y = adjacent / hypotenuse

• sin y = opposite / hypotenuse

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

• -secant^2 y * dy/dx

• secant^2 y * dy/dx

• -cosecant^2 y * dy/dx

• cosecant^2 y * dy/dx

How is dy/dx expressed in terms of sin y?

• dy/dx = sin^2 y

• dy/dx = -sin^2 y

• dy/dx = 1 / sin^2 y

• dy/dx = -1 / sin^2 y

What is the simplified form of dy/dx for y = arccotangent x?

• -1 / (1 - x^2)

• 1 / (1 - x^2)

• -1 / (1 + x^2)

• 1 / (1 + x^2)

What does the negative derivative of arccotangent x indicate about the function?

• The function is constant

• The function is increasing

• The function is oscillating

• The function is decreasing

What does it mean for a function to be monotonically decreasing?

• The function remains constant

• The function increases at a constant rate

• The function decreases at a constant rate

• The function oscillates

How does the graph of f(x) = arccotangent x verify the derivative's properties?

• The graph shows decreasing slopes

• The graph shows oscillating slopes

• The graph shows constant slopes

• The graph shows increasing slopes

What is the significance of the first derivative being always negative?

• The function is undefined

• The function is constant

• The function is always decreasing

• The function is always increasing