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

Problem 2.254 · hard Beautiful

1. \[ \frac{d}{d x} \left(x \ln{\left(3 x + 2 \right)} - x + \frac{2 \ln{\left(3 x + 2 \right)}}{3}\right) \]
   sumStart with the derivative of the entire expression.✓ Proved
2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \frac{2 \ln{\left(3 x + 2 \right)}}{3} \]
   sumApply the sum rule to separate the terms.✓ Proved
3. \[ = \frac{d}{d x} x \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \frac{2 \ln{\left(3 x + 2 \right)}}{3} - 1 \]
   derivativeDifferentiate the second term.✓ Proved
4. \[ = \frac{d}{d x} x \ln{\left(3 x + 2 \right)} + \frac{2 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} - 1 \]
   constant-multiplePull out the constant factor from the third term.✓ Proved
5. \[ = x \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} \frac{d}{d x} x + \frac{2 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} - 1 \]
   productApply the product rule to the first term.✓ Proved
6. \[ = x \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} + \frac{2 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} - 1 \]
   derivative algebraDifferentiate the first part of the product. Simplify the multiplication by 1.✓ Proved
7. \[ = \left(x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} - 1 \]
   algebraGroup the terms involving the derivative of the logarithm.✓ Proved
8. \[ = \frac{3 \left(x + \frac{2}{3}\right)}{3 x + 2} + \ln{\left(3 x + 2 \right)} - 1 \]
   chainApply the chain rule to the logarithm.✓ Proved
9. \[ = \ln{\left(3 x + 2 \right)} \]
   algebra simplify simplifyDistribute the 3 and simplify the numerator. Simplify the fraction. Final simplification.✓ Proved