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

Problem 2.1502 · hard Beautiful

1. \[ \frac{d}{d x} \left(- x + \left(x - \frac{3}{2}\right) \ln{\left(2 x - 3 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(x - \frac{3}{2}\right) \ln{\left(2 x - 3 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(x - \frac{3}{2}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \ln{\left(2 x - 3 \right)} \frac{d}{d x} \left(x - \frac{3}{2}\right) + \frac{d}{d x} \left(- x\right) \]
   productApply the product rule to the second term.✓ Proved
4. \[ = \left(x - \frac{3}{2}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \left(- \frac{d}{d x} \frac{3}{2} + \frac{d}{d x} x\right) \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- x\right) \]
   sumDistribute the logarithm over the subtraction.✓ Proved
5. \[ = \left(x - \frac{3}{2}\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- x\right) \]
   derivative constantDifferentiate the terms inside the parentheses. Simplify the constant term.✓ Proved
6. \[ = \frac{\left(x - \frac{3}{2}\right) \frac{d}{d x} \left(2 x - 3\right)}{2 x - 3} + \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- x\right) \]
   logarithmicApply the chain rule for the logarithm.✓ Proved
7. \[ = \frac{2 \left(x - \frac{3}{2}\right)}{2 x - 3} + \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- x\right) \]
   chainDifferentiate the inner function 2x - 3.✓ Proved
8. \[ = \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- x\right) + 1 \]
   algebra algebraMultiply the terms in the second part. Simplify the fraction.✓ Proved
9. \[ = \ln{\left(2 x - 3 \right)} \]
   derivative simplifyDifferentiate the first term. Combine the constants.✓ Proved