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

Problem 2.1516 · hard Beautiful

1. \[ \frac{d}{d x} \left(- x + \left(x + \frac{1}{3}\right) \ln{\left(3 x + 1 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(x + \frac{1}{3}\right) \ln{\left(3 x + 1 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(x + \frac{1}{3}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} \frac{d}{d x} \left(x + \frac{1}{3}\right) + \frac{d}{d x} \left(- x\right) \]
   productApply the product rule to the second term.✓ Proved
4. \[ = \left(x + \frac{1}{3}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \left(- x\right) \]
   derivativeDifferentiate the term (x + 1/3).✓ Proved
5. \[ = \frac{\left(x + \frac{1}{3}\right) \frac{d}{d x} \left(3 x + 1\right)}{3 x + 1} + \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \left(- x\right) \]
   chainApply the chain rule to the logarithm.✓ Proved
6. \[ = \frac{3 \left(x + \frac{1}{3}\right)}{3 x + 1} + \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \left(- x\right) \]
   derivativeDifferentiate the inner function 3*x + 1.✓ Proved
7. \[ = \frac{3 \left(x + \frac{1}{3}\right)}{3 x + 1} + \ln{\left(3 x + 1 \right)} - 1 \]
   derivativeEvaluate the derivatives of the remaining terms.✓ Proved
8. \[ = \ln{\left(3 x + 1 \right)} \]
   algebra simplify simplifyDistribute the 3 in the numerator. Simplify the fraction. Final simplification.✓ Proved