Derivative of \( \displaystyle \frac{\sqrt{9 x^{2} - 18 x + 10}}{3} \)

Problem 2.1584 · hard

1. \[ \frac{d}{d x} \frac{\sqrt{9 x^{2} - 18 x + 10}}{3} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{9 x^{2} - 18 x + 10}}{3} \]
   constant-multiple rewritePull out the constant factor 1/3. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(9 x^{2} - 18 x + 10\right)}{6 \sqrt{9 x^{2} - 18 x + 10}} \]
   chain algebraApply the chain rule. Multiply the constants.✓ Proved
4. \[ = \frac{\frac{d}{d x} 10 - \frac{d}{d x} 18 x + \frac{d}{d x} 9 x^{2}}{6 \sqrt{9 x^{2} - 18 x + 10}} \]
   sumDifferentiate the sum term by term.✓ Proved
5. \[ = \frac{18 x - 18}{6 \sqrt{9 x^{2} - 18 x + 10}} \]
   derivative algebraDifferentiate each term in the polynomial. Simplify the derivative of the polynomial.✓ Proved
6. \[ = \frac{3 x - 3}{\sqrt{9 x^{2} - 18 x + 10}} \]
   algebra simplifyMove the negative exponent to the denominator. Simplify the fraction by dividing by 6.✓ Proved