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

Problem 2.1931 · hard

1. \[ \frac{d}{d x} \left(\frac{x^{2}}{4} - \frac{x}{2} + \left(- \frac{x^{2}}{2} + x - \frac{1}{2}\right) \ln{\left(x - 1 \right)}\right) \]
   sumDifferentiate the sum term by term.✓ Proved
2. \[ = - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} + \frac{d}{d x} \left(- \frac{x^{2}}{2} + x - \frac{1}{2}\right) \ln{\left(x - 1 \right)} \]
   sumSplit the derivative into three parts.✓ Proved
3. \[ = \left(- \frac{x^{2}}{2} + x - \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x - 1 \right)} + \ln{\left(x - 1 \right)} \frac{d}{d x} \left(- \frac{x^{2}}{2} + x - \frac{1}{2}\right) - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
   productApply the product rule to the third term.✓ Proved
4. \[ = \left(1 - x\right) \ln{\left(x - 1 \right)} + \left(- \frac{x^{2}}{2} + x - \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
   derivativeDifferentiate the polynomial part of the product.✓ Proved
5. \[ = \left(1 - x\right) \ln{\left(x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} + \frac{- \frac{x^{2}}{2} + x - \frac{1}{2}}{x - 1} \]
   chainApply the chain rule to the logarithm.✓ Proved
6. \[ = \frac{x}{2} + \left(1 - x\right) \ln{\left(x - 1 \right)} - \frac{1}{2} + \frac{- \frac{x^{2}}{2} + x - \frac{1}{2}}{x - 1} \]
   derivative simplify algebraEvaluate the remaining derivatives. Simplify the first term. Factor out -1/2 from the numerator.✓ Proved
7. \[ = \left(1 - x\right) \ln{\left(x - 1 \right)} \]
   algebra algebra algebra simplifyRecognize the perfect square trinomial. Cancel the common factor (x - 1). Distribute the -1/2. Combine all remaining terms.✓ Proved