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

Problem 2.894 · hard

1. \[ \frac{d}{d x} \left(2 x \ln{\left(3 x + 2 \right)} - 2 x + \frac{4 \ln{\left(3 x + 2 \right)}}{3}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \frac{4 \ln{\left(3 x + 2 \right)}}{3} \]
   sumApply the sum rule for differentiation.✓ Proved
3. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(3 x + 2 \right)} + \frac{4 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} \]
   constant-multiplePull out the constant factor.✓ Proved
4. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(3 x + 2 \right)} + \frac{4 \frac{d}{d x} \left(3 x + 2\right)}{3 \left(3 x + 2\right)} \]
   chainApply the chain rule to the logarithm.✓ Proved
5. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(3 x + 2 \right)} + \frac{4}{3 x + 2} \]
   derivative algebraDifferentiate the inner function 3*x + 2. Simplify the constant terms.✓ Proved
6. \[ = 2 x \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} \frac{d}{d x} 2 x - \frac{d}{d x} 2 x + \frac{4}{3 x + 2} \]
   productApply the product rule to the first term.✓ Proved
7. \[ = \frac{2 x \frac{d}{d x} \left(3 x + 2\right)}{3 x + 2} + 2 \ln{\left(3 x + 2 \right)} - 2 + \frac{4}{3 x + 2} \]
   derivativeDifferentiate the components of the product rule.✓ Proved
8. \[ = \frac{6 x}{3 x + 2} + 2 \ln{\left(3 x + 2 \right)} - 2 + \frac{4}{3 x + 2} \]
   derivative algebraDifferentiate the inner function 3*x + 2. Multiply the terms in the second part.✓ Proved
9. \[ = 2 \ln{\left(3 x + 2 \right)} - 2 + \frac{6 x + 4}{3 x + 2} \]
   algebra algebraCombine the fractions with a common denominator. Factor the numerator of the fraction.✓ Proved
10. \[ = 2 \ln{\left(3 x + 2 \right)} \]
    simplify simplifyCancel the common term in the fraction. Final simplification.✓ Proved