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

Problem 2.544 · hard Beautiful

1. \[ \frac{d}{d x} \left(x \ln{\left(2 x - 3 \right)} - x - \frac{3 \ln{\left(2 x - 3 \right)}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - \frac{d}{d x} \frac{3 \ln{\left(2 x - 3 \right)}}{2} \]
   sumApply the sum rule to separate the terms.✓ Proved
3. \[ = \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - \frac{d}{d x} \frac{3 \ln{\left(2 x - 3 \right)}}{2} - 1 \]
   constantThe derivative of x is 1.✓ Proved
4. \[ = \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - \frac{3 \frac{d}{d x} \ln{\left(2 x - 3 \right)}}{2} - 1 \]
   constant-multiplePull out the constant factor 3/2.✓ Proved
5. \[ = \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - 1 - \frac{3 \frac{d}{d x} \left(2 x - 3\right)}{2 \left(2 x - 3\right)} \]
   chainApply the chain rule to the logarithm term.✓ Proved
6. \[ = \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - 1 - \frac{3}{2 x - 3} \]
   derivative algebraThe derivative of 2*x - 3 is 2. Simplify the product of the constants.✓ Proved
7. \[ = x \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \ln{\left(2 x - 3 \right)} \frac{d}{d x} x - 1 - \frac{3}{2 x - 3} \]
   productApply the product rule to the first term.✓ Proved
8. \[ = x \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \ln{\left(2 x - 3 \right)} - 1 - \frac{3}{2 x - 3} \]
   derivativeThe derivative of x is 1.✓ Proved
9. \[ = \frac{x \frac{d}{d x} \left(2 x - 3\right)}{2 x - 3} + \ln{\left(2 x - 3 \right)} - 1 - \frac{3}{2 x - 3} \]
   chainApply the chain rule to the logarithm term again.✓ Proved
10. \[ = \frac{2 x}{2 x - 3} + \ln{\left(2 x - 3 \right)} - 1 - \frac{3}{2 x - 3} \]
    derivative algebraThe derivative of 2*x - 3 is 2. Simplify the term with x.✓ Proved
11. \[ = \ln{\left(2 x - 3 \right)} \]
    algebra algebra simplifyCombine the fractions with the same denominator. Simplify the fraction (2x-3)/(2x-3). Final simplification.✓ Proved