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TS EAMCET · Maths · Inverse Trigonometric Functions

If \(f(x)=\sqrt{\operatorname{Cos}^{-1} \sqrt{1-x^2}}\), then \(f^{\prime}\left(\frac{1}{2}\right)=\)

  1. A \(\sqrt{\frac{2}{\pi}}\)
  2. B \(\sqrt{\frac{\pi}{2}}\)
  3. C \(-\sqrt{\frac{2}{\pi}}\)
  4. D \(-\sqrt{\frac{\pi}{2}}\)
Verified Solution

Answer & Solution

Correct Answer

(A) \(\sqrt{\frac{2}{\pi}}\)

Step-by-step Solution

Detailed explanation

\(f(x)=\sqrt{\operatorname{Sin}^{-1} x}\) \(f'(x) = \frac{1}{2\sqrt{\operatorname{Sin}^{-1} x}\sqrt{1-x^2}}\) \(f'\left(\frac{1}{2}\right) = \frac{1}{2\sqrt{\operatorname{Sin}^{-1} \left(\frac{1}{2}\right)}\sqrt{1-\left(\frac{1}{2}\right)^2}}\)…