Derivative of \( \displaystyle - \frac{x^{2}}{2} - \frac{x}{2} + \left(x^{2} + x + \frac{1}{4}\right) \ln{\left(2 x + 1 \right)} \)

Problem 2.1756 · hard

1. \[ \frac{d}{d x} \left(- \frac{x^{2}}{2} - \frac{x}{2} + \left(x^{2} + x + \frac{1}{4}\right) \ln{\left(2 x + 1 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \left(x^{2} + x + \frac{1}{4}\right) \ln{\left(2 x + 1 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(x^{2} + x + \frac{1}{4}\right) \frac{d}{d x} \ln{\left(2 x + 1 \right)} + \ln{\left(2 x + 1 \right)} \frac{d}{d x} \left(x^{2} + x + \frac{1}{4}\right) + \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) \]
   productApply the product rule to the third term.✓ Proved
4. \[ = \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} + \left(x^{2} + x + \frac{1}{4}\right) \frac{d}{d x} \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) \]
   derivativeDifferentiate the polynomial part of the product.✓ Proved
5. \[ = \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{\left(x^{2} + x + \frac{1}{4}\right) \frac{d}{d x} \left(2 x + 1\right)}{2 x + 1} \]
   chainApply the chain rule to the logarithm.✓ Proved
6. \[ = \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{2 \left(x^{2} + x + \frac{1}{4}\right)}{2 x + 1} \]
   derivativeDifferentiate the inner function of the logarithm.✓ Proved
7. \[ = - x + \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} - \frac{1}{2} + \frac{2 \left(x^{2} + x + \frac{1}{4}\right)}{2 x + 1} \]
   derivative algebraEvaluate the derivatives of the first two terms. Combine the first two terms.✓ Proved
8. \[ = - x + \frac{2 \left(x + \frac{1}{2}\right)^{2}}{2 x + 1} + \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} - \frac{1}{2} \]
   algebra algebraRewrite the quadratic term as a square. Factor the denominator.✓ Proved
9. \[ = \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} \]
   algebra algebra simplifySimplify the fraction. Simplify the remaining term. Combine all terms to get the final result.✓ Proved