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

Problem 2.1922 · hard

1. \[ \frac{d}{d x} \left(- \frac{x^{2}}{4} - x + \left(\frac{x^{2}}{2} + 2 x + 2\right) \ln{\left(x + 2 \right)}\right) \]
   sumStart with the derivative of the entire function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- \frac{x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{x^{2}}{2} + 2 x + 2\right) \ln{\left(x + 2 \right)} \]
   sumSplit the derivative into parts using the sum rule.✓ Proved
3. \[ = \left(\frac{x^{2}}{2} + 2 x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \ln{\left(x + 2 \right)} \frac{d}{d x} \left(\frac{x^{2}}{2} + 2 x + 2\right) + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- \frac{x^{2}}{4}\right) \]
   productApply the product rule to the third term.✓ Proved
4. \[ = \left(x + 2\right) \ln{\left(x + 2 \right)} + \left(\frac{x^{2}}{2} + 2 x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- \frac{x^{2}}{4}\right) \]
   derivativeDifferentiate the polynomial part of the product.✓ Proved
5. \[ = \left(x + 2\right) \ln{\left(x + 2 \right)} + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- \frac{x^{2}}{4}\right) + \frac{\frac{x^{2}}{2} + 2 x + 2}{x + 2} \]
   derivativeDifferentiate the logarithmic part using the chain rule.✓ Proved
6. \[ = - \frac{x}{2} + \left(x + 2\right) \ln{\left(x + 2 \right)} - 1 + \frac{\frac{x^{2}}{2} + 2 x + 2}{x + 2} \]
   derivativeDifferentiate the remaining simple terms.✓ Proved
7. \[ = - \frac{x}{2} + \left(x + 2\right) \ln{\left(x + 2 \right)} - 1 + \frac{0.5 x^{2} + 2 x + 2}{x + 2} \]
   algebraConvert coefficients to decimals for consistency.✓ Proved
8. \[ = \left(x + 2\right) \ln{\left(x + 2 \right)} \]
   algebra algebra algebra simplifyFactor the quadratic expression in the numerator. Simplify the fraction by canceling (x + 2). Distribute the constant. Combine all remaining terms.✓ Proved