Question

Multiple Choice Which of the following is \(\frac{d}{d x} \sin ^{-1}\left(\frac{x}{2}\right) ?\) \((\mathbf{A})-\frac{2}{\sqrt{4-x^{2}}} \quad(\mathbf{B})-\frac{1}{\sqrt{4-x^{2}}} \quad\) (C) \(\frac{2}{4+x^{2}}\) (D) \(\frac{2}{\sqrt{4-x^{2}}} \quad\) (E) \(\frac{1}{\sqrt{4-x^{2}}}\)

Short answer

The answer is (D) \(2 \div \sqrt{4 - x^2}\).

Step-by-step solution

Identify \(u\) and find its derivative

Here \(u = \frac{x}{2}\). Hence, \(\frac{du}{dx} = \frac{1}{2}\).

Apply the derivative rule for inverse sine

We have known the general derivative rule for inverse sine: \(\frac{d}{dx}\sin^{-1}(u) = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx}\). So, we substitute \(u = \frac{x}{2}\) into this equation: \(\frac{d}{dx}\sin^{-1}\left(\frac{x}{2}\right) = \frac{1}{\sqrt{1 - \left(\frac{x}{2}\right)^2}} \cdot \frac{1}{2}\).

Simplify the result

Simplify the result to get the final answer: \(\frac{d}{dx}\sin^{-1}\left(\frac{x}{2}\right) = 2 \div \sqrt{4 - x^2}\).