Derivative of \( \displaystyle - \frac{x^{2}}{4} + \frac{3 x}{2} + \left(\frac{x^{2}}{2} - 3 x + \frac{9}{2}\right) \ln{\left(x - 3 \right)} \)

Problem 2.1577 · hard

1. \[ \frac{d}{d x} \left(- \frac{x^{2}}{4} + \frac{3 x}{2} + \left(\frac{x^{2}}{2} - 3 x + \frac{9}{2}\right) \ln{\left(x - 3 \right)}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{x^{2}}{2} - 3 x + \frac{9}{2}\right) \ln{\left(x - 3 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(\frac{x^{2}}{2} - 3 x + \frac{9}{2}\right) \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} \left(\frac{x^{2}}{2} - 3 x + \frac{9}{2}\right) + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{4}\right) \]
   productApply the product rule to the third term.✓ Proved
4. \[ = \left(x - 3\right) \ln{\left(x - 3 \right)} + \left(\frac{x^{2}}{2} - 3 x + \frac{9}{2}\right) \frac{d}{d x} \ln{\left(x - 3 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{4}\right) \]
   derivative algebraDifferentiate the polynomial part of the product. Simplify the polynomial term.✓ Proved
5. \[ = \left(x - 3\right) \ln{\left(x - 3 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{4}\right) + \frac{\frac{x^{2}}{2} - 3 x + \frac{9}{2}}{x - 3} \]
   derivativeDifferentiate the logarithm.✓ Proved
6. \[ = - \frac{x}{2} + \left(x - 3\right) \ln{\left(x - 3 \right)} + \frac{3}{2} + \frac{\frac{x^{2}}{2} - 3 x + \frac{9}{2}}{x - 3} \]
   derivative algebraEvaluate the derivatives of the power and linear terms. Simplify the first term.✓ Proved
7. \[ = - \frac{x}{2} + \left(x - 3\right) \ln{\left(x - 3 \right)} + \frac{3}{2} + \frac{0.5 x^{2} - 3 x + 4.5}{x - 3} \]
   algebraRewrite coefficients for clarity.✓ Proved
8. \[ = - \frac{x}{2} + \left(x - 3\right) \ln{\left(x - 3 \right)} + \frac{3}{2} + \frac{0.5 x^{2} - 3.0 x + 4.5}{x - 3} \]
   algebraFactor out 0.5 from the numerator.✓ Proved
9. \[ = \left(x - 3\right) \ln{\left(x - 3 \right)} \]
   algebra algebra algebra algebra algebra simplifyFactor the quadratic expression. Cancel the common factor (x - 3). Distribute the 0.5. Convert 1.5 to a fraction. Group like terms. Combine all terms to get the final result.✓ Proved