Understanding the Derivative of Arctangent Function

The video tutorial provides a detailed proof of the derivative of the arctangent function with respect to x, showing that it equals 1/(1 + x^2). The proof begins by defining y as arctangent x and using a right triangle to model the relationship between x and y. Trigonometric identities are applied to express sine and cosine in terms of x. Implicit differentiation is then used to derive the formula for the derivative. The tutorial concludes by verifying the result and analyzing the graph of the arctangent function, highlighting its monotonically increasing nature.

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

• x^2 / (1 + x^2)

• 1 / (1 - x^2)

• 1 / (1 + x^2)

• x / (1 + x^2)

In which quadrants does the angle y lie when y = arctangent x?

• First and fourth quadrants

• Third and fourth quadrants

• First and second quadrants

• Second and third quadrants

What is the relationship between tangent y and x in the triangle model?

• Tangent y = 1/x

• Tangent y = x

• Tangent y = 1 - x

• Tangent y = x^2

How is the hypotenuse of the triangle determined in the proof?

• Square root of (1 + x^2)

• Square root of (1 - x^2)

• 1 + x

• x^2 - 1

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

• Secant^2 y * dy/dx

• Tangent y * dy/dx

• Cosine y * dy/dx

• Sine y * dy/dx

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

• 1 / secant^2 y

• 1 / cosine^2 y

• 1 / sine^2 y

• 1 / tangent^2 y

What is the final derivative formula obtained in the proof?

• dy/dx = x^2 / (1 + x^2)

• dy/dx = 1 / (1 - x^2)

• dy/dx = x / (1 + x^2)

• dy/dx = 1 / (1 + x^2)

What does the positive derivative of arctangent x indicate about its graph?

• The function is constant

• The function is oscillating

• The function is decreasing

• The function is increasing

How does the slope of the tangent line behave for the arctangent function?

• It is always positive

• It is always zero

• It varies randomly

• It is always negative

What can be said about the arctangent function over its entire domain?

• It is constant

• It is monotonically increasing

• It is monotonically decreasing

• It is undefined