Derivative of \( \displaystyle \frac{\sqrt{4 \left(2 x + 1\right)^{2} + 1}}{4} \)

Problem 2.42 · medium

1. \[ \frac{d}{d x} \frac{\sqrt{4 \left(2 x + 1\right)^{2} + 1}}{4} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{4 \left(2 x + 1\right)^{2} + 1}}{4} \]
   constant-multiple rewritePull out the constant factor 1/4. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(4 \left(2 x + 1\right)^{2} + 1\right)}{8 \sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   chainApply the chain rule.✓ Proved
4. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} 4 \left(2 x + 1\right)^{2}}{8 \sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   sumApply the sum rule to the inner derivative.✓ Proved
5. \[ = \frac{\frac{d}{d x} 4 \left(2 x + 1\right)^{2}}{8 \sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   constant algebraThe derivative of the constant 1 is 0. Simplify the expression.✓ Proved
6. \[ = \frac{\frac{d}{d x} \left(2 x + 1\right)^{2}}{2 \sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   constant-multiple algebraPull out the constant factor 4 from the derivative. Simplify the coefficients (4/8 = 1/2).✓ Proved
7. \[ = \frac{\left(4 x + 2\right) \frac{d}{d x} \left(2 x + 1\right)}{2 \sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   chainApply the chain rule to the power function.✓ Proved
8. \[ = \frac{8 x + 4}{2 \sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   derivative algebraThe derivative of 2x + 1 is 2. Multiply the constants.✓ Proved
9. \[ = \frac{4 x + 2}{\sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   algebra rewrite algebraSimplify the coefficients (1/2 * 4 = 2). Rewrite the negative exponent as a denominator. Distribute the 2 into the parentheses.✓ Proved