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

Problem 2.1742 · hard

1. \[ \frac{d}{d x} \left(- 3 x + \left(3 x - \frac{9}{2}\right) \ln{\left(2 x - 3 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x - \frac{9}{2}\right) \ln{\left(2 x - 3 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(3 x - \frac{9}{2}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \ln{\left(2 x - 3 \right)} \frac{d}{d x} \left(3 x - \frac{9}{2}\right) + \frac{d}{d x} \left(- 3 x\right) \]
   productApply the product rule to the second term.✓ Proved
4. \[ = \left(3 x - \frac{9}{2}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} - \ln{\left(2 x - 3 \right)} \frac{d}{d x} \frac{9}{2} + \ln{\left(2 x - 3 \right)} \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 3 x\right) \]
   sumDistribute the derivative over the subtraction.✓ Proved
5. \[ = \left(3 x - \frac{9}{2}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} - \ln{\left(2 x - 3 \right)} \frac{d}{d x} \frac{9}{2} + 3 \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- 3 x\right) \]
   constant-multipleDifferentiate the term 3*x.✓ Proved
6. \[ = \left(3 x - \frac{9}{2}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} + 3 \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- 3 x\right) \]
   constant algebraThe derivative of a constant is zero. Simplify the expression.✓ Proved
7. \[ = 3 \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{\left(3 x - \frac{9}{2}\right) \frac{d}{d x} \left(2 x - 3\right)}{2 x - 3} \]
   chainApply the chain rule to the logarithm.✓ Proved
8. \[ = 3 \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{2 \left(3 x - \frac{9}{2}\right)}{2 x - 3} \]
   derivative algebraDifferentiate the inner function 2*x - 3. Simplify the product.✓ Proved
9. \[ = 3 \ln{\left(2 x - 3 \right)} - 3 + \frac{2 \left(3 x - \frac{9}{2}\right)}{2 x - 3} \]
   derivativeDifferentiate the remaining term -3*x.✓ Proved
10. \[ = 3 \ln{\left(2 x - 3 \right)} - 3 + \frac{6 x - 9}{2 x - 3} \]
    algebra algebraMultiply the numerator and denominator by 2. Factor out a 3 from the numerator.✓ Proved
11. \[ = 3 \ln{\left(2 x - 3 \right)} \]
    simplify algebraCancel the common factor (2*x - 3). Combine the constant terms.✓ Proved