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

Problem 2.1428 · hard

1. \[ \frac{d}{d x} \left(- \frac{x^{2}}{2} + x \left(x + 4\right) \ln{\left(x + 2 \right)} - 2 x + 4 \ln{\left(x + 2 \right)}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x \left(x + 4\right) \ln{\left(x + 2 \right)} + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x \left(x + 4\right) \ln{\left(x + 2 \right)} + 4 \frac{d}{d x} \ln{\left(x + 2 \right)} \]
   constantApply the constant multiple rule to the last term.✓ Proved
4. \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x \left(x + 4\right) \ln{\left(x + 2 \right)} + \frac{4 \frac{d}{d x} \left(x + 2\right)}{x + 2} \]
   chainApply the chain rule to the log term.✓ Proved
5. \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x \left(x + 4\right) \ln{\left(x + 2 \right)} + \frac{4}{x + 2} \]
   derivative algebraDifferentiate the inner function x + 2. Simplify the last term.✓ Proved
6. \[ = x \left(x + 4\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + x \ln{\left(x + 2 \right)} \frac{d}{d x} \left(x + 4\right) + \left(x + 4\right) \ln{\left(x + 2 \right)} \frac{d}{d x} x + \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{4}{x + 2} \]
   productApply the product rule to the second term.✓ Proved
7. \[ = x \left(x + 4\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + x \ln{\left(x + 2 \right)} \frac{d}{d x} \left(x + 4\right) + \left(x + 4\right) \ln{\left(x + 2 \right)} + \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{4}{x + 2} \]
   derivativeDifferentiate x.✓ Proved
8. \[ = x \left(x + 4\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + x \ln{\left(x + 2 \right)} + \left(x + 4\right) \ln{\left(x + 2 \right)} + \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{4}{x + 2} \]
   derivativeDifferentiate x + 4.✓ Proved
9. \[ = x \ln{\left(x + 2 \right)} + \frac{x \left(x + 4\right)}{x + 2} + \left(x + 4\right) \ln{\left(x + 2 \right)} + \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{4}{x + 2} \]
   derivative algebra algebraDifferentiate log(x + 2). Simplify the fraction. Remove unnecessary parentheses.✓ Proved
10. \[ = x \ln{\left(x + 2 \right)} + \frac{x \left(x + 4\right)}{x + 2} + \left(x + 4\right) \ln{\left(x + 2 \right)} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) - 2 + \frac{4}{x + 2} \]
    derivativeDifferentiate -2x.✓ Proved
11. \[ = x \ln{\left(x + 2 \right)} - x + \frac{x \left(x + 4\right)}{x + 2} + \left(x + 4\right) \ln{\left(x + 2 \right)} - 2 + \frac{4}{x + 2} \]
    derivativeDifferentiate -x**2/2.✓ Proved
12. \[ = - x + \left(2 x + 4\right) \ln{\left(x + 2 \right)} - 2 + \frac{x^{2} + 4 x}{x + 2} + \frac{4}{x + 2} \]
    algebraCombine log terms.✓ Proved
13. \[ = - x + \left(2 x + 4\right) \ln{\left(x + 2 \right)} - 2 + \frac{x^{2} + 4 x + 4}{x + 2} \]
    algebraCombine the rational terms.✓ Proved
14. \[ = \left(2 x + 4\right) \ln{\left(x + 2 \right)} \]
    algebra algebra simplifyFactor the numerator. Simplify the fraction. Combine like terms.✓ Proved