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

Problem 2.1776 · hard

1. \[ \frac{d}{d x} \frac{2 \sqrt{9 x^{2} + 6 x + 2}}{3} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{2 \frac{d}{d x} \sqrt{9 x^{2} + 6 x + 2}}{3} \]
   constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(9 x^{2} + 6 x + 2\right)}{3 \sqrt{9 x^{2} + 6 x + 2}} \]
   chain algebraApply the chain rule. Simplify the constant coefficients.✓ Proved
4. \[ = \frac{18 x + 6}{3 \sqrt{9 x^{2} + 6 x + 2}} \]
   derivativeDifferentiate the polynomial inside.✓ Proved
5. \[ = \frac{2 \left(3 x + 1\right)}{\sqrt{9 x^{2} + 6 x + 2}} \]
   algebraFactor out 6 from the derivative.✓ Proved
6. \[ = \frac{6 x + 2}{\sqrt{9 x^{2} + 6 x + 2}} \]
   algebra algebra rewrite simplifySimplify the product of constants. Move the power to the denominator. Rewrite the fractional power as a square root. Factor out 2 from the numerator.✓ Proved