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

Problem 2.244 · medium

1. \[ \frac{d}{d x} \sqrt{\left(2 x - 1\right)^{2} + 1} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(\left(2 x - 1\right)^{2} + 1\right)}{2 \sqrt{\left(2 x - 1\right)^{2} + 1}} \]
   chainApply the chain rule for the square root function.✓ Proved
3. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \left(2 x - 1\right)^{2}}{2 \sqrt{\left(2 x - 1\right)^{2} + 1}} \]
   sumApply the sum rule to the inner expression.✓ Proved
4. \[ = \frac{\left(4 x - 2\right) \frac{d}{d x} \left(2 x - 1\right)}{2 \sqrt{\left(2 x - 1\right)^{2} + 1}} \]
   powerApply the power rule and chain rule to the squared term.✓ Proved
5. \[ = \frac{8 x - 4}{2 \sqrt{\left(2 x - 1\right)^{2} + 1}} \]
   derivative algebra algebraDifferentiate the linear term. Multiply the constants in the numerator. Combine the terms into a single fraction.✓ Proved
6. \[ = \frac{4 x - 2}{\sqrt{\left(2 x - 1\right)^{2} + 1}} \]
   simplifySimplify the fraction by canceling the common factor of 2.✓ Proved
7. \[ = \frac{4 x - 2}{\sqrt{4 x^{2} - 4 x + 2}} \]
   algebra simplifyExpand the squared binomial inside the square root. Combine the constant terms inside the square root.✓ Proved
8. \[ = \frac{\sqrt{2} \left(4 x - 2\right)}{2 \sqrt{2 x^{2} - 2 x + 1}} \]
   algebraFactor out a 2 from the expression inside the square root.✓ Proved
9. \[ = \frac{\sqrt{2} \left(2 x - 1\right)}{\sqrt{2 x^{2} - 2 x + 1}} \]
   algebraWait, let's simplify the coefficient instead.✓ Proved
10. \[ = \frac{2 \left(2 x - 1\right)}{\sqrt{4 x^{2} - 4 x + 2}} \]
    algebraIncorrect logic in previous thought, let's use a clean simplification.✓ Proved
11. \[ = \frac{\sqrt{2} \left(2 x - 1\right)}{\sqrt{2 x^{2} - 2 x + 1}} \]
    algebra algebraLet's re-evaluate the simplification of (2/sqrt(2)). Simplify the constant factor.✓ Proved