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

Problem 2.1491 · hard Beautiful

1. \[ \frac{d}{d x} \left(x \operatorname{asin}{\left(x \right)} + \sqrt{1 - x^{2}}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x \operatorname{asin}{\left(x \right)} + \frac{d}{d x} \sqrt{1 - x^{2}} \]
   sum rewriteApply the sum rule. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{d}{d x} x \operatorname{asin}{\left(x \right)} + \frac{\frac{d}{d x} \left(1 - x^{2}\right)}{2 \sqrt{1 - x^{2}}} \]
   chainApply the chain rule to the second term.✓ Proved
4. \[ = \frac{d}{d x} x \operatorname{asin}{\left(x \right)} + \frac{\frac{d}{d x} \left(- x^{2}\right)}{2 \sqrt{1 - x^{2}}} \]
   constantApply the sum rule to the inner expression.✓ Proved
5. \[ = - \frac{x}{\sqrt{1 - x^{2}}} + \frac{d}{d x} x \operatorname{asin}{\left(x \right)} \]
   power algebraDifferentiate the term -x**2. Simplify the constants.✓ Proved
6. \[ = x \frac{d}{d x} \operatorname{asin}{\left(x \right)} - \frac{x}{\sqrt{1 - x^{2}}} + \operatorname{asin}{\left(x \right)} \frac{d}{d x} x \]
   productApply the product rule to the first term.✓ Proved
7. \[ = x \frac{d}{d x} \operatorname{asin}{\left(x \right)} - \frac{x}{\sqrt{1 - x^{2}}} + \operatorname{asin}{\left(x \right)} \]
   derivativeDifferentiate x.✓ Proved
8. \[ = \operatorname{asin}{\left(x \right)} \]
   inverse-trig algebraDifferentiate asin(x). The two remaining terms cancel out.✓ Proved