Derivatives of Inverse Trigonometric Functions

The video tutorial explains how to find the derivatives of inverse trigonometric functions using the chain rule. It covers two examples: the derivative of the inverse sine of 3x and the inverse cosine of 4x cubed. The tutorial highlights the importance of identifying the inner function and applying the derivative formulas correctly, noting the negative sign in the inverse cosine derivative. Simplification steps are also demonstrated for clarity.

What is the inner function u when finding the derivative of f(x) = inverse sine of 3x?

• sin(x)

• 4x^3

• x^3

• 3x

What is the derivative of the inner function u = 3x?

• 1

• 3

• x

• 0

Which formula is used to find the derivative of inverse sine?

• 1 / (1 + u^2) * u'

• -1 / (1 + u^2) * u'

• 1 / sqrt(1 - u^2) * u'

• -1 / sqrt(1 - u^2) * u'

What is the simplified form of f'(x) for f(x) = inverse sine of 3x?

• 3 / sqrt(1 - 9x^2)

• -3 / sqrt(1 + 9x^2)

• 3 / sqrt(1 + 9x^2)

• -3 / sqrt(1 - 9x^2)

What is the inner function u when finding the derivative of g(x) = inverse cosine of 4x^3?

• cos(x)

• 4x^3

• x^2

• 3x

What is the derivative of the inner function u = 4x^3?

• x^3

• 3x^2

• 4x^2

• 12x^2

What is the key difference in the derivative formulas for inverse sine and inverse cosine?

• Inverse sine has a positive sign

• Inverse cosine has a negative sign

• Inverse cosine has a positive sign

• Inverse sine has a negative sign

Which formula is used to find the derivative of inverse cosine?

• -1 / sqrt(1 - u^2) * u'

• 1 / (1 + u^2) * u'

• 1 / sqrt(1 - u^2) * u'

• -1 / (1 + u^2) * u'

What is the simplified form of g'(x) for g(x) = inverse cosine of 4x^3?

• -12x^2 / sqrt(1 + 16x^6)

• 12x^2 / sqrt(1 - 16x^6)

• -12x^2 / sqrt(1 - 16x^6)

• 12x^2 / sqrt(1 + 16x^6)

What is the value of u^2 when u = 4x^3?

• 16x^6

• 8x^3

• 4x^6

• 12x^2