If $f(x) = \cos^{-1} \sqrt{x}, 0 < x < 1$, which of the following is equal to $f'(x)$?

$\frac{-1}{2\sqrt{x(1-x)}}$

The correct answer is Option (4) → $\frac{-1}{2\sqrt{x(1-x)}}$ ##

$f(x) = \cos^{-1} \sqrt{x}$.

On differentiating, we get:

$f'(x) = \frac{-1}{\sqrt{1-(\sqrt{x})^2}} \frac{d}{dx}\sqrt{x}$

$= \frac{-1}{\sqrt{1-x}} \cdot \frac{1}{2\sqrt{x}}$

$= \frac{-1}{2\sqrt{x(1-x)}}$