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

Problem 2.1827 · hard

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. \[ = \frac{d}{d x} x - \frac{d}{d x} \left(x + 1\right) \ln{\left(x + 1 \right)} \]
   sumApply the difference rule.✓ Proved
3. \[ = - \left(x + 1\right) \frac{d}{d x} \ln{\left(x + 1 \right)} - \ln{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right) + \frac{d}{d x} x \]
   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 first part of the product rule.✓ Proved
6. \[ = - \ln{\left(x + 1 \right)} \]
   derivative algebra algebra simplifyDifferentiate the second part of the product rule. Simplify the fraction. Distribute the negative sign. Combine like terms.✓ Proved