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

Problem 2.886 · medium

1. \[ \frac{d}{d x} \left(- \frac{\sqrt{4 \left(2 x + 1\right)^{2} + 1}}{4}\right) \]
   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. Rewrite the square root as a fractional power.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \left(4 \left(2 x + 1\right)^{2} + 1\right)}{8 \sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   chain algebraApply the chain rule. Simplify the constant coefficient.✓ Proved
4. \[ = - \frac{\frac{d}{d x} 1 + \frac{d}{d x} 4 \left(2 x + 1\right)^{2}}{8 \sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   sumDifferentiate the sum inside the parenthesis.✓ Proved
5. \[ = - \frac{\frac{d}{d x} 4 \left(2 x + 1\right)^{2}}{8 \sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   constantThe derivative of a constant is zero.✓ Proved
6. \[ = - \frac{\frac{d}{d x} \left(2 x + 1\right)^{2}}{2 \sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   constant-multiplePull out the constant 4.✓ Proved
7. \[ = - \frac{\left(16 x + 8\right) \frac{d}{d x} \left(2 x + 1\right)}{8 \sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   powerApply the power rule to (2x+1)^2.✓ Proved
8. \[ = - \frac{32 x + 16}{8 \sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   chain algebraApply the chain rule to 2x + 1. Multiply the constants together.✓ Proved
9. \[ = \frac{- 4 x - 2}{\sqrt{4 \left(2 x + 1\right)^{2} + 1}} \]
   simplifySimplify the final expression.✓ Proved