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

Problem 2.1772 · hard

1. \[ \frac{d}{d x} \left(- 3 x + \left(3 x + 1\right) \ln{\left(3 x + 1 \right)}\right) \]
   sumStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x + 1\right) \ln{\left(3 x + 1 \right)} \]
   sumSplit the derivative into two parts.✓ Proved
3. \[ = \left(3 x + 1\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} \frac{d}{d x} \left(3 x + 1\right) + \frac{d}{d x} \left(- 3 x\right) \]
   productApply the product rule to the second term.✓ Proved
4. \[ = \ln{\left(3 x + 1 \right)} \frac{d}{d x} \left(3 x + 1\right) + \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x + 1\right) \]
   chain algebraApply the chain rule to the logarithm. Simplify the fraction.✓ Proved
5. \[ = 3 \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + 3 \]
   derivativeDifferentiate the linear terms.✓ Proved
6. \[ = 3 \ln{\left(3 x + 1 \right)} \]
   derivative simplifyDifferentiate the first term. Combine the constant terms.✓ Proved