Understanding the Derivative of Arc Secant Function

The video tutorial provides a detailed proof of the derivative of the function f(x) = arc secant x. It begins by setting up the problem and using a reference triangle to understand the relationships between secant, cosine, and angles. The tutorial then employs implicit differentiation to derive the expression for the derivative. The process includes simplifying the expression and ensuring the result is always positive. Finally, the video verifies the derivative graphically, showing that the function is always increasing over its domain.

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

• 1 divided by x squared minus one

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

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

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

In the reference triangle for arc secant, what does the hypotenuse represent?

• The opposite side

• The adjacent side

• The absolute value of x

• The angle y

Why must we be careful when setting up the reference triangle for arc secant?

• Because the adjacent side can be zero

• Because the opposite side can be negative

• Because the angle y can be in different quadrants

• Because the hypotenuse can be negative

How is sine y expressed using the reference triangle?

• The square root of x squared minus one divided by the absolute value of x

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

• The absolute value of x divided by the square root of x squared minus one

• The square root of x squared minus one divided by x

What is the reciprocal of secant y in terms of cosine y?

• 1 over x

• The square root of x squared minus one

• x

• The absolute value of x

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

• Secant y times tangent y times dy/dx

• Tangent y times dy/dx

• Secant y times dy/dx

• Dy/dx

How do we express dy/dx after implicit differentiation?

• 1 divided by cosine y

• 1 divided by secant y

• 1 divided by secant y times tangent y

• 1 divided by tangent y

What is the final expression for dy/dx after simplification?

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

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

• Cosine y times cotangent y

• Cosine y divided by sine y

What does the derivative of arc secant x indicate about the function's behavior?

• The function is undefined

• The function is always increasing

• The function is constant

• The function is decreasing

How is the increasing nature of the arc secant function verified?

• By calculating the derivative

• By using a reference triangle

• By solving an equation

• By graphically showing the slope of the tangent line