Derivative of \( \displaystyle 2 x \ln{\left(x + 1 \right)} - 2 x + 2 \ln{\left(x + 1 \right)} \)

Problem 2.1129 · hard Beautiful

1. \[ \frac{d}{d x} \left(2 x \ln{\left(x + 1 \right)} - 2 x + 2 \ln{\left(x + 1 \right)}\right) \]
   Differentiate the function with respect to x.✓ Proved
2. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(x + 1 \right)} + \frac{d}{d x} 2 \ln{\left(x + 1 \right)} \]
   sumApply the sum rule for differentiation.✓ Proved
3. \[ = \frac{d}{d x} 2 x \ln{\left(x + 1 \right)} + \frac{d}{d x} 2 \ln{\left(x + 1 \right)} - 2 \]
   constantThe derivative of 2*x is 2.✓ Proved
4. \[ = 2 \frac{d}{d x} x \ln{\left(x + 1 \right)} + 2 \frac{d}{d x} \ln{\left(x + 1 \right)} - 2 \]
   constant-multipleFactor out the constant 2 from the terms.✓ Proved
5. \[ = 2 x \frac{d}{d x} \ln{\left(x + 1 \right)} + 2 \ln{\left(x + 1 \right)} \frac{d}{d x} x + 2 \frac{d}{d x} \ln{\left(x + 1 \right)} - 2 \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 2 x \frac{d}{d x} \ln{\left(x + 1 \right)} + 2 \ln{\left(x + 1 \right)} + 2 \frac{d}{d x} \ln{\left(x + 1 \right)} - 2 \]
   derivativeThe derivative of x is 1.✓ Proved
7. \[ = \frac{2 x}{x + 1} + 2 \ln{\left(x + 1 \right)} - 2 + \frac{2}{x + 1} \]
   derivative algebraThe derivative of log(x + 1) is 1/(x + 1). Distribute the 2.✓ Proved
8. \[ = 2 \ln{\left(x + 1 \right)} - 2 + \frac{2 x + 2}{x + 1} \]
   algebra algebraCombine the fractions with the same denominator. Factor the numerator of the fraction.✓ Proved
9. \[ = 2 \ln{\left(x + 1 \right)} \]
   algebra simplifySimplify the fraction by canceling (x + 1). Simplify the remaining terms.✓ Proved