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

Problem 2.1173 · hard

1. \[ \frac{d}{d x} \left(2 x \ln{\left(2 x + 1 \right)} - 2 x + \ln{\left(2 x + 1 \right)}\right) \]
   Differentiate the function with respect to x.✓ 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. \[ = \frac{4 x}{2 x + 1} + 2 \ln{\left(2 x + 1 \right)} - 2 + \frac{2}{2 x + 1} \]
   logarithmic algebra algebraApply the chain rule to the log terms. Simplify the derivatives of the log terms. Multiply the terms.✓ 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 simplifyCancel the common factor in the fraction. Simplify the final expression.✓ Proved