Derivative of \( \displaystyle x \ln{\left(x - 1 \right)} - x - \ln{\left(x - 1 \right)} \)

Problem 2.963 · hard Beautiful

1. \[ \frac{d}{d x} \left(x \ln{\left(x - 1 \right)} - x - \ln{\left(x - 1 \right)}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(x - 1 \right)} - \frac{d}{d x} \ln{\left(x - 1 \right)} \]
   sumApply the sum rule for differentiation.✓ Proved
3. \[ = \frac{d}{d x} x \ln{\left(x - 1 \right)} - \frac{d}{d x} \ln{\left(x - 1 \right)} - 1 \]
   constantThe derivative of x is 1.✓ Proved
4. \[ = x \frac{d}{d x} \ln{\left(x - 1 \right)} + \ln{\left(x - 1 \right)} \frac{d}{d x} x - \frac{d}{d x} \ln{\left(x - 1 \right)} - 1 \]
   productApply the product rule to the first term.✓ Proved
5. \[ = x \frac{d}{d x} \ln{\left(x - 1 \right)} + \ln{\left(x - 1 \right)} - \frac{d}{d x} \ln{\left(x - 1 \right)} - 1 \]
   constantThe derivative of x is 1.✓ Proved
6. \[ = \frac{x}{x - 1} + \ln{\left(x - 1 \right)} - 1 - \frac{1}{x - 1} \]
   logarithmic algebraApply the derivative rule for the logarithm. Simplify the expression.✓ Proved
7. \[ = \ln{\left(x - 1 \right)} \]
   algebra algebra simplifyCombine the fractions with the same denominator. Simplify the fraction. Final simplification.✓ Proved