Derivative of \( \displaystyle 3 x \ln{\left(4 x + 2 \right)} - 3 x + \frac{3 \ln{\left(2 x + 1 \right)}}{2} \)

Problem 2.1431 · hard

1. \[ \frac{d}{d x} \left(3 x \ln{\left(4 x + 2 \right)} - 3 x + \frac{3 \ln{\left(2 x + 1 \right)}}{2}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 3 x + \frac{d}{d x} 3 x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} 3 x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   constantThe derivative of 3*x is 3.✓ Proved
4. \[ = 3 x \frac{d}{d x} \ln{\left(4 x + 2 \right)} + \ln{\left(4 x + 2 \right)} \frac{d}{d x} 3 x + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   productApply the product rule to the first term.✓ Proved
5. \[ = 3 x \frac{d}{d x} \ln{\left(4 x + 2 \right)} + 3 \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   constant-multipleThe derivative of 3*x is 3.✓ Proved
6. \[ = \frac{3 x \frac{d}{d x} \left(4 x + 2\right)}{4 x + 2} + 3 \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   chainApply the chain rule to the log term.✓ Proved
7. \[ = \frac{12 x}{4 x + 2} + 3 \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{3 \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   derivativeThe derivative of 4*x + 2 is 4.✓ Proved
8. \[ = \frac{12 x}{4 x + 2} + 3 \ln{\left(4 x + 2 \right)} + \frac{3 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} - 3 \]
   constant-multipleFactor out the constant 3/2.✓ Proved
9. \[ = \frac{12 x}{4 x + 2} + 3 \ln{\left(4 x + 2 \right)} - 3 + \frac{3 \frac{d}{d x} \left(2 x + 1\right)}{2 \left(2 x + 1\right)} \]
   chainApply the chain rule to the second log term.✓ Proved
10. \[ = \frac{12 x}{4 x + 2} + 3 \ln{\left(4 x + 2 \right)} - 3 + \frac{3}{2 x + 1} \]
    derivative algebraThe derivative of 2*x + 1 is 2. Simplify the products.✓ Proved
11. \[ = \frac{6 x}{2 x + 1} + 3 \ln{\left(4 x + 2 \right)} - 3 + \frac{3}{2 x + 1} \]
    algebraSimplify the fraction 12*x/(4*x + 2) by dividing numerator and denominator by 2.✓ Proved
12. \[ = 3 \ln{\left(4 x + 2 \right)} - 3 + \frac{6 x + 3}{2 x + 1} \]
    algebra algebraCombine the fractions with the same denominator. Factor the numerator of the fraction.✓ Proved
13. \[ = 3 \ln{\left(4 x + 2 \right)} \]
    algebra simplifySimplify the fraction (2*x + 1)/(2*x + 1) to 1. Final simplification.✓ Proved