Derivative of \( \displaystyle \sqrt{1 - x^{2}} + \operatorname{asin}{\left(x \right)} \)

Problem 2.1890 · hard

1. \[ \frac{d}{d x} \left(\sqrt{1 - x^{2}} + \operatorname{asin}{\left(x \right)}\right) \]
   sumStart with the derivative of the sum.✓ Proved
2. \[ = \frac{d}{d x} \sqrt{1 - x^{2}} + \frac{d}{d x} \operatorname{asin}{\left(x \right)} \]
   sum rewriteApply the sum rule. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \frac{d}{d x} \operatorname{asin}{\left(x \right)} + \frac{\frac{d}{d x} \left(1 - x^{2}\right)}{2 \sqrt{1 - x^{2}}} \]
   chainApply the chain rule to the first term.✓ Proved
4. \[ = \frac{d}{d x} \operatorname{asin}{\left(x \right)} + \frac{\frac{d}{d x} \left(- x^{2}\right)}{2 \sqrt{1 - x^{2}}} \]
   constantSeparate the constant term.✓ Proved
5. \[ = - \frac{x}{\sqrt{1 - x^{2}}} + \frac{d}{d x} \operatorname{asin}{\left(x \right)} \]
   power algebraDifferentiate the power term. Simplify the coefficients.✓ Proved
6. \[ = - \frac{x}{\sqrt{1 - x^{2}}} + \frac{1}{\sqrt{1 - x^{2}}} \]
   derivative algebraDifferentiate the inverse sine term. Rewrite the negative exponent.✓ Proved
7. \[ = \frac{1 - x}{\sqrt{1 - x^{2}}} \]
   simplifyCombine the fractions.✓ Proved