Derivative of \( \displaystyle \frac{\sqrt{16 x^{2} + 8 x + 2}}{4} \)

Problem 2.1520 · hard

1. \[ \frac{d}{d x} \frac{\sqrt{16 x^{2} + 8 x + 2}}{4} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{16 x^{2} + 8 x + 2}}{4} \]
   constant-multiple rewritePull out the constant factor 1/4. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(16 x^{2} + 8 x + 2\right)}{8 \sqrt{16 x^{2} + 8 x + 2}} \]
   chain algebraApply the chain rule. Multiply the constants.✓ Proved
4. \[ = \frac{32 x + 8}{8 \sqrt{16 x^{2} + 8 x + 2}} \]
   derivativeDifferentiate the polynomial inside.✓ Proved
5. \[ = \frac{4 x + 1}{\sqrt{16 x^{2} + 8 x + 2}} \]
   algebra simplifyFactor out 8 from the polynomial. Simplify the expression.✓ Proved