Derivative of \( \displaystyle - \frac{\sqrt{\left(3 x + 1\right)^{2} + 1}}{3} \)

Problem 2.629 · medium

1. \[ \frac{d}{d x} \left(- \frac{\sqrt{\left(3 x + 1\right)^{2} + 1}}{3}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{\frac{d}{d x} \sqrt{\left(3 x + 1\right)^{2} + 1}}{3} \]
   constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional exponent.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \left(\left(3 x + 1\right)^{2} + 1\right)}{6 \sqrt{\left(3 x + 1\right)^{2} + 1}} \]
   chainApply the chain rule.✓ Proved
4. \[ = - \frac{\frac{d}{d x} \left(3 x + 1\right)^{2}}{6 \sqrt{\left(3 x + 1\right)^{2} + 1}} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
5. \[ = - \frac{\left(6 x + 2\right) \frac{d}{d x} \left(3 x + 1\right)}{6 \sqrt{\left(3 x + 1\right)^{2} + 1}} \]
   powerApply the power rule to the squared term.✓ Proved
6. \[ = - \frac{18 x + 6}{6 \sqrt{\left(3 x + 1\right)^{2} + 1}} \]
   derivative constant-multipleDifferentiate the linear term. Multiply the constants.✓ Proved
7. \[ = \frac{- 3 x - 1}{\sqrt{\left(3 x + 1\right)^{2} + 1}} \]
   simplifySimplify the expression.✓ Proved