Derivative of \( \displaystyle - \frac{x^{2}}{2} + 3 x + \left(x^{2} - 6 x + 9\right) \ln{\left(x - 3 \right)} \)

Problem 2.2072 · hard

1. \[ \frac{d}{d x} \left(- \frac{x^{2}}{2} + 3 x + \left(x^{2} - 6 x + 9\right) \ln{\left(x - 3 \right)}\right) \]
   sumStart with the derivative of the entire function.✓ Proved
2. \[ = \frac{d}{d x} 3 x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \left(x^{2} - 6 x + 9\right) \ln{\left(x - 3 \right)} \]
   sumApply the sum rule to separate the terms.✓ Proved
3. \[ = \left(x^{2} - 6 x + 9\right) \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} \left(x^{2} - 6 x + 9\right) + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) \]
   productApply the product rule to the third term.✓ Proved
4. \[ = \left(2 x - 6\right) \ln{\left(x - 3 \right)} + \left(x^{2} - 6 x + 9\right) \frac{d}{d x} \ln{\left(x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) \]
   derivativeDifferentiate the polynomial part of the product.✓ Proved
5. \[ = \left(2 x - 6\right) \ln{\left(x - 3 \right)} + \frac{d}{d x} 3 x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{x^{2} - 6 x + 9}{x - 3} \]
   derivativeDifferentiate the logarithmic part.✓ Proved
6. \[ = - x + \left(2 x - 6\right) \ln{\left(x - 3 \right)} + 3 + \frac{x^{2} - 6 x + 9}{x - 3} \]
   derivativeEvaluate the derivatives of the first two terms.✓ Proved
7. \[ = \left(2 x - 6\right) \ln{\left(x - 3 \right)} \]
   algebra simplify algebra simplifyRecognize that x**2 - 6*x + 9 is (x - 3)**2. Simplify the fraction. Expand the simplified term. Combine all terms to get the final result.✓ Proved