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

Problem 2.354 · hard Beautiful

1. \[ \frac{d}{d x} \left(x \ln{\left(2 x + 2 \right)} - x + \ln{\left(x + 1 \right)}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(2 x + 2 \right)} + \frac{d}{d x} \ln{\left(x + 1 \right)} \]
   sumApply the sum rule to differentiate each term separately.✓ Proved
3. \[ = \frac{d}{d x} x \ln{\left(2 x + 2 \right)} + \frac{d}{d x} \ln{\left(x + 1 \right)} - 1 \]
   constantThe derivative of x is 1.✓ Proved
4. \[ = x \frac{d}{d x} \ln{\left(2 x + 2 \right)} + \ln{\left(2 x + 2 \right)} \frac{d}{d x} x + \frac{d}{d x} \ln{\left(x + 1 \right)} - 1 \]
   productApply the product rule to the first term.✓ Proved
5. \[ = x \frac{d}{d x} \ln{\left(2 x + 2 \right)} + \ln{\left(2 x + 2 \right)} + \frac{d}{d x} \ln{\left(x + 1 \right)} - 1 \]
   constantThe derivative of x is 1.✓ Proved
6. \[ = \frac{x \frac{d}{d x} \left(2 x + 2\right)}{2 x + 2} + \ln{\left(2 x + 2 \right)} - 1 + \frac{\frac{d}{d x} \left(x + 1\right)}{x + 1} \]
   chainApply the chain rule to the logarithmic terms.✓ Proved
7. \[ = \frac{2 x}{2 x + 2} + \ln{\left(2 x + 2 \right)} - 1 + \frac{1}{x + 1} \]
   derivative algebraDifferentiate the inner functions. Simplify the products.✓ Proved
8. \[ = \frac{x}{x + 1} + \ln{\left(2 x + 2 \right)} - 1 + \frac{1}{x + 1} \]
   algebraSimplify the fraction (2*x)/(2*x + 2) by dividing numerator and denominator by 2.✓ Proved
9. \[ = \ln{\left(2 x + 2 \right)} \]
   algebra algebra simplifyCombine the fractions with the same denominator. Simplify the fraction (x + 1)/(x + 1) to 1. Combine the remaining terms.✓ Proved