Understanding the Derivative of Arccosine

The video tutorial provides a step-by-step proof of the derivative of the arccosine function. It begins by setting up the problem and using a right triangle to establish trigonometric relationships. Implicit differentiation is applied to find the derivative dy/dx, which is shown to be negative one divided by the square root of one minus x squared. The tutorial analyzes the properties of the derivative, noting where it is defined and its behavior. Finally, the derivative's behavior is verified graphically, showing that the function is decreasing where the derivative is defined.

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

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

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

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

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

In the context of arccosine, what does the angle y represent?

• The cosine function value

• The sine function value

• The tangent function value

• The angle corresponding to the cosine value

How is the opposite side of the triangle labeled in terms of x?

• The square root of x squared

• x squared

• 1 minus x squared

• The square root of 1 minus x squared

What is the range of the angle y in the context of arccosine?

• From 0 to 2pi

• From -pi/2 to pi/2

• From -pi to pi

• From 0 to pi

What is the result of differentiating cosine y with respect to x?

• Cosine y times dy/dx

• Negative cosine y times dy/dx

• Sine y times dy/dx

• Negative sine y times dy/dx

What is the value of dy/dx after solving the implicit differentiation?

• 1 divided by sine y

• Negative 1 divided by sine y

• 1 divided by cosine y

• Negative 1 divided by cosine y

Why is the derivative of arccosine x not defined at x equals 1 or -1?

• Because the function is not continuous

• Because the function is not differentiable

• Because the function is not integrable

• Because it results in division by zero

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

• The function is oscillating

• The function is decreasing

• The function is constant

• The function is increasing

What is the significance of the derivative being negative where it is defined?

• The tangent lines are vertical

• The tangent lines have negative slopes

• The tangent lines have positive slopes

• The tangent lines are horizontal

What does the graph of f(x) = arccosine x show about the function?

• The function is increasing where the derivative is defined

• The function is oscillating where the derivative is defined

• The function is decreasing where the derivative is defined

• The function is constant where the derivative is defined