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

Problem 2.169 · medium

1. \[ \frac{d}{d x} \left(- \frac{\sqrt{\left(2 x - 1\right)^{2} + 1}}{2}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{\frac{d}{d x} \sqrt{\left(2 x - 1\right)^{2} + 1}}{2} \]
   constant rewritePull out the constant factor -1/2. Rewrite the square root as a fractional power.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \left(\left(2 x - 1\right)^{2} + 1\right)}{4 \sqrt{\left(2 x - 1\right)^{2} + 1}} \]
   chain algebraApply the chain rule. Multiply the constants.✓ Proved
4. \[ = - \frac{\left(4 x - 2\right) \frac{d}{d x} \left(2 x - 1\right)}{4 \sqrt{\left(2 x - 1\right)^{2} + 1}} \]
   powerApply the power rule to the inner squared term.✓ Proved
5. \[ = - \frac{8 x - 4}{4 \sqrt{\left(2 x - 1\right)^{2} + 1}} \]
   derivative algebraDifferentiate the innermost linear term. Simplify the product of constants.✓ Proved
6. \[ = \frac{1 - 2 x}{\sqrt{\left(2 x - 1\right)^{2} + 1}} \]
   simplifySimplify the final expression by combining terms.✓ Proved