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

Problem 2.1743 · hard

1. \[ \frac{d}{d x} \left(\frac{x^{2}}{2} - \frac{x}{2} + \left(- x^{2} + x - \frac{1}{4}\right) \ln{\left(2 x - 1 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{d}{d x} \left(- x^{2} + x - \frac{1}{4}\right) \ln{\left(2 x - 1 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(- x^{2} + x - \frac{1}{4}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \ln{\left(2 x - 1 \right)} \frac{d}{d x} \left(- x^{2} + x - \frac{1}{4}\right) - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} \]
   productApply the product rule to the third term.✓ Proved
4. \[ = \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} + \left(- x^{2} + x - \frac{1}{4}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} \]
   derivativeDifferentiate the polynomial part.✓ Proved
5. \[ = \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{\left(- x^{2} + x - \frac{1}{4}\right) \frac{d}{d x} \left(2 x - 1\right)}{2 x - 1} \]
   chainApply the chain rule to the logarithm.✓ Proved
6. \[ = \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{2 \left(- x^{2} + x - \frac{1}{4}\right)}{2 x - 1} \]
   derivativeDifferentiate the inner function 2x - 1.✓ Proved
7. \[ = \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{- 2 x^{2} + 2 x - \frac{1}{2}}{2 x - 1} \]
   algebraSimplify the fraction.✓ Proved
8. \[ = x + \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} + \frac{- 2 x^{2} + 2 x - \frac{1}{2}}{2 x - 1} \]
   derivative algebraDifferentiate the first two terms. Distribute the 2 in the numerator.✓ Proved
9. \[ = x + \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} + \frac{- 4 x^{2} + 4 x - 1}{4 x - 2} \]
   algebraMultiply numerator and denominator by 2 to clear the fraction.✓ Proved
10. \[ = x + \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} - \frac{4 x^{2} - 4 x + 1}{4 x - 2} \]
    algebraFactor out the negative sign.✓ Proved
11. \[ = x + \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{\left(2 x - 1\right)^{2}}{4 x - 2} - \frac{1}{2} \]
    algebraRecognize the perfect square trinomial.✓ Proved
12. \[ = \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} \]
    algebra algebra simplifyCancel the common factor (2*x - 1). Split the fraction. Combine all terms to get the final answer.✓ Proved