Derivative of \( \displaystyle \sqrt{x^{2} - 2 x + 2} \)

Problem 2.1479 · hard

1. \[ \frac{d}{d x} \sqrt{x^{2} - 2 x + 2} \]
   derivative rewriteStart with the derivative of the function. Rewrite the square root as a fractional power.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(x^{2} - 2 x + 2\right)}{2 \sqrt{x^{2} - 2 x + 2}} \]
   chainApply the chain rule.✓ Proved
3. \[ = \frac{\frac{d}{d x} 2 - \frac{d}{d x} 2 x + \frac{d}{d x} x^{2}}{2 \sqrt{x^{2} - 2 x + 2}} \]
   sumDifferentiate the terms inside the parentheses separately.✓ Proved
4. \[ = \frac{2 x - 2}{2 \sqrt{x^{2} - 2 x + 2}} \]
   derivative algebraDifferentiate each term. Simplify the terms inside the parentheses.✓ Proved
5. \[ = \frac{x - 1}{\sqrt{x^{2} - 2 x + 2}} \]
   constant-multiple simplifyDistribute the 1/2 into the parentheses. Rewrite the expression in a standard form.✓ Proved