Derivative of \( \displaystyle 2 x \ln{\left(2 x - 1 \right)} - 2 x - \ln{\left(2 x - 1 \right)} \)

Problem 2.1309 · hard

1. \[ \frac{d}{d x} \left(2 x \ln{\left(2 x - 1 \right)} - 2 x - \ln{\left(2 x - 1 \right)}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \ln{\left(2 x - 1 \right)} \]
   sumApply the sum rule for differentiation.✓ Proved
3. \[ = \frac{d}{d x} 2 x \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \ln{\left(2 x - 1 \right)} - 2 \]
   constantThe derivative of 2*x is 2.✓ Proved
4. \[ = 2 x \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \ln{\left(2 x - 1 \right)} \frac{d}{d x} 2 x - \frac{d}{d x} \ln{\left(2 x - 1 \right)} - 2 \]
   productApply the product rule to the first term.✓ Proved
5. \[ = 2 x \frac{d}{d x} \ln{\left(2 x - 1 \right)} + 2 \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \ln{\left(2 x - 1 \right)} - 2 \]
   constant-multipleThe derivative of 2*x is 2.✓ Proved
6. \[ = \frac{4 x}{2 x - 1} + 2 \ln{\left(2 x - 1 \right)} - 2 - \frac{2}{2 x - 1} \]
   chain algebraApply the chain rule to the log terms. Simplify the chain rule results.✓ Proved
7. \[ = 2 \ln{\left(2 x - 1 \right)} - 2 + \frac{4 x - 2}{2 x - 1} \]
   algebra algebraCombine the fractions. Factor the numerator.✓ Proved
8. \[ = 2 \ln{\left(2 x - 1 \right)} \]
   simplify simplifySimplify the expression. Final simplified result.✓ Proved