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

Problem 2.1064 · hard

1. \[ \frac{d}{d x} \left(- \frac{x^{2}}{2} + x \left(x - 6\right) \ln{\left(x - 3 \right)} + 3 x + 9 \ln{\left(x - 3 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} 3 x + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x \left(x - 6\right) \ln{\left(x - 3 \right)} + \frac{d}{d x} 9 \ln{\left(x - 3 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x \left(x - 6\right) \ln{\left(x - 3 \right)} + \frac{d}{d x} 9 \ln{\left(x - 3 \right)} + 3 \]
   constantDifferentiate the linear term 3*x.✓ Proved
4. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x \left(x - 6\right) \ln{\left(x - 3 \right)} + 9 \frac{d}{d x} \ln{\left(x - 3 \right)} + 3 \]
   constant-multiplePull out the constant 9.✓ Proved
5. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x \left(x - 6\right) \ln{\left(x - 3 \right)} + 3 + \frac{9 \frac{d}{d x} \left(x - 3\right)}{x - 3} \]
   chainApply the chain rule to log(x - 3).✓ Proved
6. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x \left(x - 6\right) \ln{\left(x - 3 \right)} + 3 + \frac{9}{x - 3} \]
   derivative algebraDifferentiate x - 3. Simplify the expression.✓ Proved
7. \[ = - x + \frac{d}{d x} x \left(x - 6\right) \ln{\left(x - 3 \right)} + 3 + \frac{9}{x - 3} \]
   derivativeDifferentiate -x**2/2.✓ Proved
8. \[ = x \left(x - 6\right) \frac{d}{d x} \ln{\left(x - 3 \right)} - x + \ln{\left(x - 3 \right)} \frac{d}{d x} x \left(x - 6\right) + 3 + \frac{9}{x - 3} \]
   productApply the product rule to x*(x - 6)*log(x - 3).✓ Proved
9. \[ = x \left(x - 6\right) \frac{d}{d x} \ln{\left(x - 3 \right)} - x + \left(x \frac{d}{d x} \left(x - 6\right) + \left(x - 6\right) \frac{d}{d x} x\right) \ln{\left(x - 3 \right)} + 3 + \frac{9}{x - 3} \]
   productApply the product rule to (x*(x - 6))*log(x - 3).✓ Proved
10. \[ = x \left(x - 6\right) \frac{d}{d x} \ln{\left(x - 3 \right)} - x + \left(2 x - 6\right) \ln{\left(x - 3 \right)} + 3 + \frac{9}{x - 3} \]
    derivative algebra algebraDifferentiate the components of the product rule. Simplify the expression inside the parentheses. Combine like terms.✓ Proved
11. \[ = \frac{x \left(x - 6\right)}{x - 3} - x + \left(2 x - 6\right) \ln{\left(x - 3 \right)} + 3 + \frac{9}{x - 3} \]
    derivativeDifferentiate log(x - 3).✓ Proved
12. \[ = - x + \left(2 x - 6\right) \ln{\left(x - 3 \right)} + 3 + \frac{x^{2} - 6 x}{x - 3} + \frac{9}{x - 3} \]
    algebraMultiply the terms.✓ Proved
13. \[ = - x + \left(2 x - 6\right) \ln{\left(x - 3 \right)} + \frac{x^{2} - 3 x}{x - 3} \]
    algebra algebra algebraFind a common denominator for the rational terms. Expand the numerator. Simplify the numerator.✓ Proved
14. \[ = \left(2 x - 6\right) \ln{\left(x - 3 \right)} \]
    algebra algebra algebraFactor the numerator. Cancel the common factor (x - 3). Final simplification.✓ Proved