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

Problem 2.1600 · hard

1. \[ \frac{d}{d x} \left(- \frac{x^{2}}{2} - 2 x + \left(x^{2} + 4 x + 4\right) \ln{\left(x + 2 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \left(x^{2} + 4 x + 4\right) \ln{\left(x + 2 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(x^{2} + 4 x + 4\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \ln{\left(x + 2 \right)} \frac{d}{d x} \left(x^{2} + 4 x + 4\right) - \frac{d}{d x} 2 x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) \]
   productApply the product rule to the third term.✓ Proved
4. \[ = \left(2 x + 4\right) \ln{\left(x + 2 \right)} + \left(x^{2} + 4 x + 4\right) \frac{d}{d x} \ln{\left(x + 2 \right)} - \frac{d}{d x} 2 x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) \]
   derivativeDifferentiate the polynomial term.✓ Proved
5. \[ = \left(2 x + 4\right) \ln{\left(x + 2 \right)} - \frac{d}{d x} 2 x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{x^{2} + 4 x + 4}{x + 2} \]
   derivativeDifferentiate the logarithm term.✓ Proved
6. \[ = x + \left(2 x + 4\right) \ln{\left(x + 2 \right)} - \frac{d}{d x} 2 x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + 2 \]
   algebra algebraFactor the quadratic term. Simplify the fraction.✓ Proved
7. \[ = \left(2 x + 4\right) \ln{\left(x + 2 \right)} \]
   derivative simplifyCompute the derivatives of the remaining terms. Combine like terms.✓ Proved