Derivative of \( \displaystyle \sqrt{x^{2} - 6 x + 10} \)

Problem 2.1493 · hard

1. \[ \frac{d}{d x} \sqrt{x^{2} - 6 x + 10} \]
   derivative rewriteStart with the derivative of the function. Rewrite the square root as a fractional power.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(x^{2} - 6 x + 10\right)}{2 \sqrt{x^{2} - 6 x + 10}} \]
   chainApply the chain rule.✓ Proved
3. \[ = \frac{\frac{d}{d x} 10}{2 \sqrt{x^{2} - 6 x + 10}} - \frac{\frac{d}{d x} 6 x}{2 \sqrt{x^{2} - 6 x + 10}} + \frac{\frac{d}{d x} x^{2}}{2 \sqrt{x^{2} - 6 x + 10}} \]
   sumDifferentiate the sum term by term.✓ Proved
4. \[ = \frac{x}{\sqrt{x^{2} - 6 x + 10}} - \frac{3}{\sqrt{x^{2} - 6 x + 10}} \]
   derivative algebra constant-multipleDifferentiate each term. The derivative of the constant 10 is 0. Simplify the coefficients.✓ Proved
5. \[ = \frac{x - 3}{\sqrt{x^{2} - 6 x + 10}} \]
   algebra simplifyFactor out the common term. Rewrite with a positive exponent and radical form.✓ Proved