Derivative of \( \displaystyle - 3 x + \left(3 x + 6\right) \ln{\left(x + 2 \right)} \)

Problem 2.2122 · hard Beautiful

1. \[ \frac{d}{d x} \left(- 3 x + \left(3 x + 6\right) \ln{\left(x + 2 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x + 6\right) \ln{\left(x + 2 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(3 x + 6\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \ln{\left(x + 2 \right)} \frac{d}{d x} \left(3 x + 6\right) + \frac{d}{d x} \left(- 3 x\right) \]
   productApply the product rule to the second term.✓ Proved
4. \[ = \left(3 x + 6\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \ln{\left(x + 2 \right)} \frac{d}{d x} \left(3 x + 6\right) - 3 \]
   constantDifferentiate the first term.✓ Proved
5. \[ = \left(3 x + 6\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + 3 \ln{\left(x + 2 \right)} - 3 \]
   derivativeDifferentiate the term (3*x + 6).✓ Proved
6. \[ = 3 \ln{\left(x + 2 \right)} - 3 + \frac{3 x + 6}{x + 2} \]
   derivative algebraDifferentiate the log term. Factor out 3 from the numerator.✓ Proved
7. \[ = 3 \ln{\left(x + 2 \right)} \]
   simplify simplifySimplify the fraction. Combine the constant terms.✓ Proved