Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Which of the following represents the derivative of sin⁻¹(u)?

1. 1/√(1-u²) du/dx

2. 1/(1-u²) du/dx

3. 1/√(1+u²) du/dx

4. 1/u√(1-u²) du/dx

The correct answer is: 1/√(1-u²) du/dx

The derivative of the inverse sine function, sin⁻¹(u), can be derived using implicit differentiation or by referring to standard derivative formulas. The correct representation of this derivative is 1/√(1-u²) du/dx. This reflects the chain rule of differentiation, which indicates that when taking the derivative of a composite function, you also need to multiply by the derivative of the inner function (in this case, du/dx). The term √(1-u²) serves as the normalization factor that accounts for the range of the inverse sine function. To further clarify why this is the correct choice, the derivative 1/√(1-u²) arises specifically because the inverse sine function has a domain restricted to the interval [-1, 1]. Thus, the term 1-u² ensures that we remain within this boundary when calculating the derivative, as it reflects the characteristics of the unit circle (where u represents sin(θ)). The other options do not conform to the established derivative for sin⁻¹(u). For instance, the option involving 1/(1-u²) lacks the necessary square root component and does not correspond to the geometric foundation of the sine function. Similarly, options with 1/√