Derivative of \( \displaystyle 5 x \ln{\left(2 x - 1 \right)} - 5 x - \frac{5 \ln{\left(2 x - 1 \right)}}{2} \)

Problem 2.51 · hard

1. \[ \frac{d}{d x} \left(5 x \ln{\left(2 x - 1 \right)} - 5 x - \frac{5 \ln{\left(2 x - 1 \right)}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 5 x + \frac{d}{d x} 5 x \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{5 \ln{\left(2 x - 1 \right)}}{2} \]
   sumApply the sum rule to separate the terms.✓ Proved
3. \[ = - \frac{d}{d x} 5 x + \frac{d}{d x} 5 x \ln{\left(2 x - 1 \right)} - \frac{5 \frac{d}{d x} \ln{\left(2 x - 1 \right)}}{2} \]
   constant-multiplePull out the constant factor from the third term.✓ Proved
4. \[ = - \frac{d}{d x} 5 x + 5 \frac{d}{d x} x \ln{\left(2 x - 1 \right)} - \frac{5 \frac{d}{d x} \ln{\left(2 x - 1 \right)}}{2} \]
   constant-multiplePull out the constant factor from the first term.✓ Proved
5. \[ = 5 x \frac{d}{d x} \ln{\left(2 x - 1 \right)} + 5 \ln{\left(2 x - 1 \right)} \frac{d}{d x} x - \frac{d}{d x} 5 x - \frac{5 \frac{d}{d x} \ln{\left(2 x - 1 \right)}}{2} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 5 x \frac{d}{d x} \ln{\left(2 x - 1 \right)} + 5 \ln{\left(2 x - 1 \right)} - \frac{d}{d x} 5 x - \frac{5 \frac{d}{d x} \ln{\left(2 x - 1 \right)}}{2} \]
   derivativeDifferentiate x.✓ Proved
7. \[ = 5 x \frac{d}{d x} \ln{\left(2 x - 1 \right)} + 5 \ln{\left(2 x - 1 \right)} - \frac{5 \frac{d}{d x} \ln{\left(2 x - 1 \right)}}{2} - 5 \]
   derivative algebraDifferentiate 5*x. Distribute the 5 into the parentheses.✓ Proved
8. \[ = \left(5 x - \frac{5}{2}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + 5 \ln{\left(2 x - 1 \right)} - 5 \]
   algebraGroup the terms involving the derivative of the logarithm.✓ Proved
9. \[ = 5 \ln{\left(2 x - 1 \right)} - 5 + \frac{\left(5 x - \frac{5}{2}\right) \frac{d}{d x} \left(2 x - 1\right)}{2 x - 1} \]
   chainApply the chain rule to the logarithm.✓ Proved
10. \[ = 5 \ln{\left(2 x - 1 \right)} - 5 + \frac{2 \left(5 x - \frac{5}{2}\right)}{2 x - 1} \]
    derivative algebraDifferentiate 2*x - 1. Simplify the expression.✓ Proved
11. \[ = 5 \ln{\left(2 x - 1 \right)} - 5 + \frac{10 x - 5}{2 x - 1} \]
    algebraMultiply the numerator and denominator.✓ Proved
12. \[ = 5 \ln{\left(2 x - 1 \right)} \]
    algebra algebra algebra simplifyCombine the terms over a common denominator. Distribute the -5 in the numerator. Simplify the numerator. Final simplification.✓ Proved