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 2.2178 · 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. \[ = - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{4} + \frac{d}{d x} \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) \ln{\left(x - 3 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
   productApply the product rule to the third term.✓ Proved
4. \[ = \left(3 - x\right) \ln{\left(x - 3 \right)} + \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) \frac{d}{d x} \ln{\left(x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
   derivativeDifferentiate the polynomial part of the product.✓ Proved
5. \[ = \left(3 - x\right) \ln{\left(x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{4} + \frac{- \frac{x^{2}}{2} + 3 x - \frac{9}{2}}{x - 3} \]
   logarithmicDifferentiate the natural logarithm.✓ Proved
6. \[ = \frac{x}{2} + \left(3 - x\right) \ln{\left(x - 3 \right)} - \frac{3}{2} + \frac{- \frac{x^{2}}{2} + 3 x - \frac{9}{2}}{x - 3} \]
   derivative simplifyDifferentiate the remaining power and constant terms. Simplify the first term.✓ Proved
7. \[ = \frac{x}{2} + \left(3 - x\right) \ln{\left(x - 3 \right)} - \frac{3}{2} + \frac{- 0.5 x^{2} + 3 x - 4.5}{x - 3} \]
   algebraRewrite coefficients as decimals for clarity.✓ Proved
8. \[ = \frac{x}{2} + \left(3 - x\right) \ln{\left(x - 3 \right)} - \frac{3}{2} + \frac{- 0.5 x^{2} + 3.0 x - 4.5}{x - 3} \]
   algebraFactor out -0.5 from the numerator.✓ Proved
9. \[ = \left(3 - x\right) \ln{\left(x - 3 \right)} \]
   algebra simplify algebra algebra simplifyFactor the quadratic expression. Cancel the common factor (x - 3). Distribute the -0.5. Rewrite 0.5*x and 1.5 as fractions. Combine all terms to find the final simplified derivative.✓ Proved