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

Problem 2.899 · medium

1. \[ \frac{d}{d x} \frac{\sqrt{\left(2 x + 1\right)^{2} + 1}}{2} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{\left(2 x + 1\right)^{2} + 1}}{2} \]
   constant-multiple rewritePull out the constant factor 1/2. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(\left(2 x + 1\right)^{2} + 1\right)}{4 \sqrt{\left(2 x + 1\right)^{2} + 1}} \]
   chainApply the chain rule.✓ Proved
4. \[ = \frac{\frac{d}{d x} \left(2 x + 1\right)^{2}}{4 \sqrt{\left(2 x + 1\right)^{2} + 1}} \]
   sumDifferentiate the sum inside the parenthesis.✓ Proved
5. \[ = \frac{\left(4 x + 2\right) \frac{d}{d x} \left(2 x + 1\right)}{4 \sqrt{\left(2 x + 1\right)^{2} + 1}} \]
   powerApply the power rule to the squared term.✓ Proved
6. \[ = \frac{8 x + 4}{4 \sqrt{\left(2 x + 1\right)^{2} + 1}} \]
   derivative algebraDifferentiate the linear term. Multiply the constants.✓ Proved
7. \[ = \frac{2 x + 1}{\sqrt{\left(2 x + 1\right)^{2} + 1}} \]
   simplifySimplify the final expression.✓ Proved