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

Problem 2.1589 · hard

1. \[ \frac{d}{d x} \left(- \frac{x^{2}}{4} - \frac{x}{2} + \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \ln{\left(x + 1 \right)}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \left(- \frac{x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \ln{\left(x + 1 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \ln{\left(x + 1 \right)} \frac{d}{d x} \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) + \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \left(- \frac{x^{2}}{4}\right) \]
   productApply the product rule to the third term.✓ Proved
4. \[ = \left(x + 1\right) \ln{\left(x + 1 \right)} + \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \left(- \frac{x^{2}}{4}\right) \]
   derivative algebraDifferentiate the polynomial part of the product. Simplify the polynomial expression.✓ Proved
5. \[ = \left(x + 1\right) \ln{\left(x + 1 \right)} + \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \left(- \frac{x^{2}}{4}\right) + \frac{\frac{x^{2}}{2} + x + \frac{1}{2}}{x + 1} \]
   derivativeDifferentiate the logarithm.✓ Proved
6. \[ = - \frac{x}{2} + \left(x + 1\right) \ln{\left(x + 1 \right)} - \frac{1}{2} + \frac{\frac{x^{2}}{2} + x + \frac{1}{2}}{x + 1} \]
   derivative algebra algebraDifferentiate the remaining power and constant terms. Simplify the first term. Factor out 1/2 from the numerator.✓ Proved
7. \[ = \left(x + 1\right) \ln{\left(x + 1 \right)} \]
   algebra algebra algebra simplifyRecognize the perfect square trinomial. Cancel the common factor (x + 1). Distribute the 1/2. Combine like terms.✓ Proved