Derivative of \( \displaystyle - 2 x + \left(2 x + 4\right) \ln{\left(x + 2 \right)} \)

Problem 2.1591 · hard Beautiful

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