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

Problem 2.410 · hard Beautiful

1. \[ \frac{d}{d x} \left(\sqrt{1 - \left(x - 1\right)^{2}} + \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(x - 1\right) \operatorname{asin}{\left(x - 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(x - 1\right)^{2}} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(x - 1\right) \frac{d}{d x} \operatorname{asin}{\left(x - 1 \right)} + \operatorname{asin}{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right) + \frac{d}{d x} \sqrt{1 - \left(x - 1\right)^{2}} \]
   productApply the product rule to the second term.✓ Proved
4. \[ = \left(x - 1\right) \frac{d}{d x} \operatorname{asin}{\left(x - 1 \right)} + \operatorname{asin}{\left(x - 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(x - 1\right)^{2}} \]
   derivative constant-multipleDifferentiate (x - 1). Simplify the constant factor.✓ Proved
5. \[ = \operatorname{asin}{\left(x - 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(x - 1\right)^{2}} + \frac{\left(x - 1\right) \frac{d}{d x} \left(x - 1\right)}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
   inverse-trigApply the derivative rule for arcsine.✓ Proved
6. \[ = \operatorname{asin}{\left(x - 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(x - 1\right)^{2}} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
   derivative algebra rewriteDifferentiate (x - 1). Simplify the fraction. Rewrite the square root as a power.✓ Proved
7. \[ = \operatorname{asin}{\left(x - 1 \right)} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} + \frac{\frac{d}{d x} \left(1 - \left(x - 1\right)^{2}\right)}{2 \sqrt{1 - \left(x - 1\right)^{2}}} \]
   chainApply the chain rule.✓ Proved
8. \[ = \operatorname{asin}{\left(x - 1 \right)} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} + \frac{\frac{d}{d x} 1 - \frac{d}{d x} \left(x - 1\right)^{2}}{2 \sqrt{1 - \left(x - 1\right)^{2}}} \]
   algebraDistribute the derivative over the subtraction.✓ Proved
9. \[ = \operatorname{asin}{\left(x - 1 \right)} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} - \frac{\frac{d}{d x} \left(x - 1\right)^{2}}{2 \sqrt{1 - \left(x - 1\right)^{2}}} \]
   derivativeDifferentiate the constant 1.✓ Proved
10. \[ = \operatorname{asin}{\left(x - 1 \right)} + \frac{2 - 2 x}{2 \sqrt{1 - \left(x - 1\right)^{2}}} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
    powerApply the power rule to (x - 1)**2.✓ Proved
11. \[ = \frac{1 - x}{\sqrt{1 - \left(x - 1\right)^{2}}} + \operatorname{asin}{\left(x - 1 \right)} + \frac{x - 1}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
    algebraSimplify the expression by multiplying terms.✓ Proved
12. \[ = \operatorname{asin}{\left(x - 1 \right)} \]
    simplifyThe two terms involving the square root cancel each other out.✓ Proved