Derivative of \( \displaystyle \operatorname{asin}{\left(\sqrt{x} \right)} \)

Problem 2.481 · hard

1. \[ \frac{d}{d x} \operatorname{asin}{\left(\sqrt{x} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{x}}{\sqrt{1 - x}} \]
   chain algebraApply the chain rule to the arcsine function. Simplify the term inside the square root.✓ Proved
3. \[ = \frac{1}{2 \sqrt{x} \sqrt{1 - x}} \]
   derivative algebraDifferentiate the inner square root function. Combine the terms into a single fraction.✓ Proved
4. \[ = \frac{1}{2 \sqrt{x \left(1 - x\right)}} \]
   simplifySimplify the expression by bringing x into the square root.≈ Checked numerically