Derivative of \( \displaystyle 3 x \ln{\left(2 x + 1 \right)} - 3 x + \frac{3 \ln{\left(2 x + 1 \right)}}{2} \)

Problem 2.994 · hard

1. \[ \frac{d}{d x} \left(3 x \ln{\left(2 x + 1 \right)} - 3 x + \frac{3 \ln{\left(2 x + 1 \right)}}{2}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 3 x + \frac{d}{d x} 3 x \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} 3 x \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   constantThe derivative of 3*x is 3.✓ Proved
4. \[ = 3 \frac{d}{d x} x \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   constant-multipleFactor out the constant 3 from the first term.✓ Proved
5. \[ = 3 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + 3 \ln{\left(2 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 3 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + 3 \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   derivativeThe derivative of x is 1.✓ Proved
7. \[ = \frac{3 x \frac{d}{d x} \left(2 x + 1\right)}{2 x + 1} + 3 \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   chainApply the chain rule to log(2*x + 1).✓ Proved
8. \[ = \frac{6 x}{2 x + 1} + 3 \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   derivative algebraThe derivative of 2*x + 1 is 2. Simplify the fraction.✓ Proved
9. \[ = \frac{6 x}{2 x + 1} + 3 \ln{\left(2 x + 1 \right)} + \frac{3 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   constant-multipleFactor out 3/2 from the last term.✓ Proved
10. \[ = \frac{6 x}{2 x + 1} + 3 \ln{\left(2 x + 1 \right)} - 3 + \frac{3}{2 x + 1} \]
    derivative algebra algebraDifferentiate the log term again. Simplify the last term. Distribute the 3.✓ Proved
11. \[ = 3 \ln{\left(2 x + 1 \right)} - 3 + \frac{6 x + 3}{2 x + 1} \]
    algebra algebraCombine the fractions. Factor the numerator.✓ Proved
12. \[ = 3 \ln{\left(2 x + 1 \right)} \]
    algebra simplifyCancel the (2*x + 1) terms. Final simplification.✓ Proved