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

Problem 2.2001 · hard Beautiful

1. \[ \frac{d}{d x} \left(- 3 x + \left(3 x + 3\right) \ln{\left(x + 1 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x + 3\right) \ln{\left(x + 1 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x \ln{\left(x + 1 \right)} + 3 \ln{\left(x + 1 \right)}\right) \]
   algebraDistribute the term.✓ Proved
4. \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} 3 x \ln{\left(x + 1 \right)} + \frac{d}{d x} 3 \ln{\left(x + 1 \right)} \]
   sumApply the sum rule again.✓ Proved
5. \[ = \frac{d}{d x} \left(- 3 x\right) + 3 \frac{d}{d x} x \ln{\left(x + 1 \right)} + 3 \frac{d}{d x} \ln{\left(x + 1 \right)} \]
   constant-multipleFactor out the constant 3.✓ Proved
6. \[ = 3 x \frac{d}{d x} \ln{\left(x + 1 \right)} + 3 \ln{\left(x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} \left(- 3 x\right) + 3 \frac{d}{d x} \ln{\left(x + 1 \right)} \]
   productApply the product rule to the middle term.✓ Proved
7. \[ = 3 x \frac{d}{d x} \ln{\left(x + 1 \right)} + 3 \ln{\left(x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + 3 \frac{d}{d x} \ln{\left(x + 1 \right)} \]
   derivative algebraDifferentiate x. Distribute the 3.✓ Proved
8. \[ = \frac{3 x}{x + 1} + 3 \ln{\left(x + 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{3}{x + 1} \]
   derivativeDifferentiate log(x + 1).✓ Proved
9. \[ = \frac{3 x}{x + 1} + 3 \ln{\left(x + 1 \right)} - 3 + \frac{3}{x + 1} \]
   derivativeDifferentiate -3*x.✓ Proved
10. \[ = 3 \ln{\left(x + 1 \right)} - 3 + \frac{3 x + 3}{x + 1} \]
    algebra algebraCombine the fractions. Factor out 3 from the numerator.✓ Proved
11. \[ = 3 \ln{\left(x + 1 \right)} \]
    algebra simplifySimplify the fraction. Combine the constants.✓ Proved