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

Problem 2.2008 · hard

1. \[ \frac{d}{d x} \left(x - \left(x + \frac{1}{2}\right) \ln{\left(2 x + 1 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x - \frac{d}{d x} \left(x + \frac{1}{2}\right) \ln{\left(2 x + 1 \right)} \]
   sumApply the difference rule.✓ Proved
3. \[ = - \left(x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(2 x + 1 \right)} - \ln{\left(2 x + 1 \right)} \frac{d}{d x} \left(x + \frac{1}{2}\right) + \frac{d}{d x} x \]
   productApply the product rule to the second term.✓ Proved
4. \[ = - \left(x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(2 x + 1 \right)} - \ln{\left(2 x + 1 \right)} \frac{d}{d x} \frac{1}{2} - \ln{\left(2 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]
   sumApply the sum rule to the first part of the product.✓ Proved
5. \[ = - \left(x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(2 x + 1 \right)} - \ln{\left(2 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]
   constant simplifyThe derivative of the constant 1/2 is 0. Simplify the expression by removing the zero term.✓ Proved
6. \[ = - \frac{\left(x + \frac{1}{2}\right) \frac{d}{d x} \left(2 x + 1\right)}{2 x + 1} - \ln{\left(2 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]
   chainApply the chain rule to the logarithm.✓ Proved
7. \[ = - \frac{2 \left(x + \frac{1}{2}\right)}{2 x + 1} - \ln{\left(2 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]
   derivativeDifferentiate the inner function 2x + 1.✓ Proved
8. \[ = - \frac{2 \left(x + \frac{1}{2}\right)}{2 x + 1} - \ln{\left(2 x + 1 \right)} + 1 \]
   derivativeEvaluate the derivatives of the basic terms.✓ Proved
9. \[ = - \ln{\left(2 x + 1 \right)} \]
   algebra algebra simplify simplifyDistribute the negative sign. Distribute the 2 in the numerator. Simplify the fraction. Combine the constant terms.✓ Proved