Understanding Derivatives of Inverse Trigonometric Functions

The video tutorial explains how to find derivatives of functions involving inverse trigonometric functions, focusing on arc sine and arc cosine. It introduces derivative formulas, emphasizing the importance of identifying composite functions to apply the chain rule correctly. Two examples are provided: one for a non-composite function using inverse sine and another for a composite function using inverse cosine, demonstrating the application of the chain rule. The tutorial concludes with a discussion on the preference for derivative formulas that include the chain rule.

What is the primary focus when finding derivatives of inverse trigonometric functions?

• Avoiding the chain rule

• Identifying composite functions

• Memorizing all derivative formulas

• Using only basic trigonometric identities

Why are there two derivative formulas for inverse sine and cosine?

• To avoid using the chain rule

• To account for different trigonometric identities

• To differentiate between basic and composite functions

• To simplify calculations

What is the role of U' in the chain rule?

• It is the derivative of the outer function

• It is the derivative of the inner function

• It is a constant

• It is the original function

In the example of F(x) = 3 * inverse sin(x), why is the chain rule not necessary?

• Because the derivative is zero

• Because the inner function is a constant

• Because the function is composite

• Because it is not a composite function

What is the derivative of F(x) = 3 * inverse sin(x)?

• 3 / sqrt(1 - x^2)

• 3x / sqrt(1 - x^2)

• 3 / (1 - x^2)

• 3x^2 / sqrt(1 - x^2)

What happens if you apply the chain rule to a non-composite function?

• The result is incorrect

• The derivative becomes zero

• The function becomes undefined

• The correct derivative is still obtained

In the example of G(x) = 4 * inverse cosine(3x^2), what is the inner function?

• x^2

• inverse cosine

• 3x^2

• 4x

What is the derivative of G(x) = 4 * inverse cosine(3x^2)?

• 4 / sqrt(1 - 9x^4)

• -4 / sqrt(1 - 9x^4)

• 4x / sqrt(1 - 9x^4)

• -4x / sqrt(1 - 9x^4)

Why might some textbooks not include the chain rule in derivative formulas?

• To simplify the learning process

• To emphasize the importance of composite functions

• To reduce the number of formulas

• To avoid confusion with basic functions

What is the benefit of using derivative formulas that include the chain rule?

• They are easier to memorize

• They highlight the need to identify composite functions

• They eliminate the need for calculations

• They are more accurate