Derivative of \( \displaystyle - \frac{\sqrt{9 x^{2} + 12 x + 5}}{3} \)

Problem 2.2191 · hard

1. \[ \frac{d}{d x} \left(- \frac{\sqrt{9 x^{2} + 12 x + 5}}{3}\right) \]
   constant-multipleStart with the derivative of the function.✓ Proved
2. \[ = - \frac{\frac{d}{d x} \sqrt{9 x^{2} + 12 x + 5}}{3} \]
   constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional power.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \left(9 x^{2} + 12 x + 5\right)}{6 \sqrt{9 x^{2} + 12 x + 5}} \]
   chain algebraApply the chain rule. Multiply the constants.✓ Proved
4. \[ = - \frac{18 x + 12}{6 \sqrt{9 x^{2} + 12 x + 5}} \]
   derivative rewriteDifferentiate the inner polynomial. Rewrite the negative exponent.✓ Proved
5. \[ = \frac{- 18 x - 12}{6 \sqrt{9 x^{2} + 12 x + 5}} \]
   algebraCombine the terms into a single fraction.✓ Proved
6. \[ = \frac{- 3 x - 2}{\sqrt{9 x^{2} + 12 x + 5}} \]
   simplifySimplify the fraction by dividing the numerator and denominator by 6.✓ Proved