Derivatives of Inverse Trigonometric Functions

This video tutorial covers the derivatives of inverse trigonometric functions, starting with an overview of derivative rules. It includes a detailed derivation of the inverse sine derivative using implicit differentiation and the Pythagorean theorem. The tutorial provides examples of calculating derivatives for arc tangent, arc secant, and arc cosine, demonstrating the application of differentiation rules. It concludes with an example of simplifying a composite function involving arc cosine before finding its derivative.

What is the relationship between the derivatives of inverse sine and inverse cosine functions?

• They are negatives of each other.

• They are unrelated.

• They are identical.

• They are reciprocals of each other.

Which method is used to derive the derivative of the inverse sine function?

• Numerical differentiation

• Explicit differentiation

• Implicit differentiation

• Graphical differentiation

In the derivation of the inverse sine function, what trigonometric identity is used to express cosine(y)?

• cos(y) = sqrt(1 - x^2)

• cos(y) = 1 + x^2

• cos(y) = x^2 - 1

• cos(y) = 1 - x^2

When finding the derivative of the inverse tangent function, what is the first step?

• Use the quotient rule

• Find the second derivative

• Identify U and find U prime

• Apply the product rule

What is the derivative of the inverse tangent function in terms of U?

• U / (1 + U^2)

• U / (1 - U^2)

• 1 / (1 - U^2)

• 1 / (1 + U^2)

In the example of the inverse secant function, what simplification is made to the expression?

• The absolute value is removed.

• The factor of five is simplified.

• The square root is expanded.

• The expression is left unsimplified.

What is the derivative of the inverse secant function in terms of U?

• 1 / (|U| * sqrt(U^2 - 1))

• U / (|U| * sqrt(U^2 - 1))

• 1 / (|U| * sqrt(1 - U^2))

• U / (|U| * sqrt(1 - U^2))

In the example involving the inverse cosine function, what rule is applied to find the derivative?

• Power rule with chain rule

• Quotient rule

• Implicit differentiation

• Product rule

What is the simplified form of the derivative of the function involving the inverse cosine?

• -t / sqrt(1 - t^2)

• t / sqrt(1 + t^2)

• -t / sqrt(1 + t^2)

• t / sqrt(1 - t^2)

What is the purpose of rewriting the function as a power before finding its derivative?

• To make it easier to integrate

• To apply the product rule

• To apply the power rule with the chain rule

• To simplify the expression