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

Problem 2.1206 · hard

1. \[ \frac{d}{d x} \left(- x \ln{\left(2 x + 1 \right)} + x - \frac{\ln{\left(2 x + 1 \right)}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x + \frac{d}{d x} \left(- x \ln{\left(2 x + 1 \right)}\right) - \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \left(- x \ln{\left(2 x + 1 \right)}\right) - \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} + 1 \]
   derivativeDifferentiate the term x.✓ Proved
4. \[ = \frac{d}{d x} \left(- x \ln{\left(2 x + 1 \right)}\right) - \frac{\frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} + 1 \]
   constant-multiplePull out the constant factor 1/2.✓ Proved
5. \[ = - x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + \ln{\left(2 x + 1 \right)} \frac{d}{d x} \left(- x\right) - \frac{\frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} + 1 \]
   productApply the product rule to the first term.✓ Proved
6. \[ = - x \frac{d}{d x} \ln{\left(2 x + 1 \right)} - \ln{\left(2 x + 1 \right)} - \frac{\frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} + 1 \]
   derivative algebraDifferentiate -x. Simplify the coefficient.✓ Proved
7. \[ = - \left(x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(2 x + 1 \right)} - \ln{\left(2 x + 1 \right)} + 1 \]
   algebraGroup the terms containing the derivative.✓ Proved
8. \[ = - \frac{\left(x + \frac{1}{2}\right) \frac{d}{d x} \left(2 x + 1\right)}{2 x + 1} - \ln{\left(2 x + 1 \right)} + 1 \]
   chainApply the chain rule to the logarithm.✓ Proved
9. \[ = - \frac{2 \left(x + \frac{1}{2}\right)}{2 x + 1} - \ln{\left(2 x + 1 \right)} + 1 \]
   derivativeDifferentiate the inner function 2*x + 1.✓ Proved
10. \[ = - \ln{\left(2 x + 1 \right)} \]
    algebra algebra algebraSimplify the product (x + 1/2) * (2 / (2x + 1)). Simplify the fraction. Final simplification.✓ Proved