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

Problem 2.296 · hard Beautiful

1. \[ \frac{d}{d x} \left(3 x \ln{\left(x + 2 \right)} - 3 x + 6 \ln{\left(x + 2 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 3 x + \frac{d}{d x} 3 x \ln{\left(x + 2 \right)} + \frac{d}{d x} 6 \ln{\left(x + 2 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} 3 x \ln{\left(x + 2 \right)} + \frac{d}{d x} 6 \ln{\left(x + 2 \right)} - 3 \]
   constantDifferentiate the linear term.✓ Proved
4. \[ = 3 x \frac{d}{d x} \ln{\left(x + 2 \right)} + 3 \ln{\left(x + 2 \right)} + \frac{d}{d x} 6 \ln{\left(x + 2 \right)} - 3 \]
   productApply the product rule to the first term.✓ Proved
5. \[ = \frac{3 x}{x + 2} + 3 \ln{\left(x + 2 \right)} - 3 + \frac{6}{x + 2} \]
   logarithmic algebraDifferentiate the logarithmic terms. Simplify the fractions.✓ Proved
6. \[ = 3 \ln{\left(x + 2 \right)} - 3 + \frac{3 x + 6}{x + 2} \]
   algebra algebraCombine the fractions with the same denominator. Factor the numerator of the fraction.✓ Proved
7. \[ = 3 \ln{\left(x + 2 \right)} \]
   algebra simplifySimplify the fraction. Final simplification.✓ Proved