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

Problem 2.1522 · hard

1. \[ \frac{d}{d x} \sqrt{9 x^{2} + 6 x + 2} \]
   derivative rewriteStart with the derivative of the function. Rewrite the square root as a fractional exponent.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(9 x^{2} + 6 x + 2\right)}{2 \sqrt{9 x^{2} + 6 x + 2}} \]
   chainApply the chain rule.✓ Proved
3. \[ = \frac{\frac{d}{d x} 2 + \frac{d}{d x} 6 x + \frac{d}{d x} 9 x^{2}}{2 \sqrt{9 x^{2} + 6 x + 2}} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
4. \[ = \frac{18 x + 6}{2 \sqrt{9 x^{2} + 6 x + 2}} \]
   derivative constantDifferentiate each term individually. The derivative of the constant 2 is 0.✓ Proved
5. \[ = \frac{9 x + 3}{\sqrt{9 x^{2} + 6 x + 2}} \]
   constant-multiple simplifyDistribute the 1/2 into the parentheses. Rewrite the negative exponent as a fraction.✓ Proved