Ohio Assessments for Educators (OAE) Mathematics Practice Exam

The derivative of cos⁻¹(u) can be expressed as?

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 cosine function, denoted as cos⁻¹(u), is derived using the chain rule from calculus. The general formula for the derivative of the inverse cosine function is given as: \[ \frac{d}{du}[\cos^{-1}(u)] = -\frac{1}{\sqrt{1 - u^2}}. \] When applying the chain rule, this derivative is multiplied by the derivative of \( u \) with respect to \( x \), which is represented as \( \frac{du}{dx} \). Thus, the full expression becomes: \[ \frac{d}{dx}[\cos^{-1}(u)] = -\frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx}. \] This matches the correct choice. The negative sign indicates that as \( u \) increases, the value of the inverse cosine function decreases, which is consistent with the properties of the cosine function. Additionally, the \( \sqrt{1-u^2} \) term stems from the Pythagorean identity, ensuring that the derivative is only defined for values \( u \) in the range of -1 to 1, where