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

Problem 2.1669 · hard Beautiful

1. \[ \frac{d}{d x} \left(- x + \left(x + 1\right) \ln{\left(x + 1 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(x + 1\right) \ln{\left(x + 1 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(x + 1\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \ln{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right) + \frac{d}{d x} \left(- x\right) \]
   productApply the product rule to the second term.✓ Proved
4. \[ = \left(x + 1\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \ln{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right) - 1 \]
   constantDifferentiate the first term.✓ Proved
5. \[ = \left(x + 1\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \ln{\left(x + 1 \right)} - 1 \]
   derivativeDifferentiate the term (x + 1).✓ Proved
6. \[ = \ln{\left(x + 1 \right)} \]
   derivative algebra simplifyDifferentiate the log term. Simplify the product. Combine like terms.✓ Proved