Derivative of \( \displaystyle - \frac{\sqrt{9 x^{2} - 6 x + 2}}{3} \)

Problem 2.1912 · hard

1. \[ \frac{d}{d x} \left(- \frac{\sqrt{9 x^{2} - 6 x + 2}}{3}\right) \]
   constant-multipleStart with the derivative of the function.✓ Proved
2. \[ = - \frac{\frac{d}{d x} \sqrt{9 x^{2} - 6 x + 2}}{3} \]
   constant-multiple rewritePull out the constant factor. Rewrite the square root as a power.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \left(9 x^{2} - 6 x + 2\right)}{6 \sqrt{9 x^{2} - 6 x + 2}} \]
   chain algebraApply the chain rule. Simplify the constant coefficient.✓ Proved
4. \[ = - \frac{\frac{d}{d x} 2 - \frac{d}{d x} 6 x + \frac{d}{d x} 9 x^{2}}{6 \sqrt{9 x^{2} - 6 x + 2}} \]
   sumDifferentiate the terms inside the parentheses separately.✓ Proved
5. \[ = - \frac{18 x - 6}{6 \sqrt{9 x^{2} - 6 x + 2}} \]
   derivative constantCalculate the derivatives of each term. The derivative of the constant 2 is 0.✓ Proved
6. \[ = \frac{1 - 3 x}{\sqrt{9 x^{2} - 6 x + 2}} \]
   algebraRearrange the terms.✓ Proved
7. \[ = \frac{6 - 18 x}{6 \sqrt{9 x^{2} - 6 x + 2}} \]
   algebraMove the negative exponent to the denominator.✓ Proved
8. \[ = \frac{1 - 3 x}{\sqrt{9 x^{2} - 6 x + 2}} \]
   simplifySimplify the fraction by dividing by 6.✓ Proved