Derivative of \( \displaystyle 3 x \ln{\left(x + 1 \right)} - 3 x + 3 \ln{\left(x + 1 \right)} \)

Problem 2.1338 · hard Beautiful

1. \[ \frac{d}{d x} \left(3 x \ln{\left(x + 1 \right)} - 3 x + 3 \ln{\left(x + 1 \right)}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 3 x + \frac{d}{d x} 3 x \ln{\left(x + 1 \right)} + \frac{d}{d x} 3 \ln{\left(x + 1 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} 3 x \ln{\left(x + 1 \right)} + \frac{d}{d x} 3 \ln{\left(x + 1 \right)} - 3 \]
   constantDifferentiate the term 3*x.✓ Proved
4. \[ = 3 \frac{d}{d x} x \ln{\left(x + 1 \right)} + 3 \frac{d}{d x} \ln{\left(x + 1 \right)} - 3 \]
   constant-multipleFactor out the constant 3 from the terms.✓ Proved
5. \[ = 3 x \frac{d}{d x} \ln{\left(x + 1 \right)} + 3 \ln{\left(x + 1 \right)} \frac{d}{d x} x + 3 \frac{d}{d x} \ln{\left(x + 1 \right)} - 3 \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 3 x \frac{d}{d x} \ln{\left(x + 1 \right)} + 3 \ln{\left(x + 1 \right)} + 3 \frac{d}{d x} \ln{\left(x + 1 \right)} - 3 \]
   derivativeDifferentiate x.✓ Proved
7. \[ = \frac{3 x}{x + 1} + 3 \ln{\left(x + 1 \right)} - 3 + \frac{3}{x + 1} \]
   derivative algebraDifferentiate log(x + 1). Distribute the 3.✓ Proved
8. \[ = 3 \ln{\left(x + 1 \right)} - 3 + \frac{3 x + 3}{x + 1} \]
   algebra algebraGroup the terms with the common denominator (x + 1). Factor out 3 from the numerator.✓ Proved
9. \[ = 3 \ln{\left(x + 1 \right)} \]
   simplify simplifySimplify the fraction. Subtract 3.✓ Proved